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Full text of "Brief relational mathematics counseling as an approach to mathematics academic support of college students taking introductory courses: a dissertation"

LUDCKE LIBRARY 
Lesley University 
30 Mellen Street 
Cambridge, MA 02138-2790 



FOR REFERENCE 

Do Not Take From This Room 



Digitized by tine Internet Arciiive 

in 2010 witii funding from 
Lesley University, Sherrill Library 



http://www.archive.org/details/briefrelationalmOOjill 



Brief Relational Mathematics Counseling 

as an Approach to Mathematics Academic Support 

of College Students taking Introductory Courses 



A DISSERTATION 



submitted by 



Jillian M. Knowles 



In partial fiilfillment of the requirements 

for degree of 

Doctor of Philosophy 



LESLEY UNIVERSITY 

May 24 
2004 



TABLE OF CONTENTS 

Page 

ACKNOWLEDGEMENTS Hi 

ABSTRACT v 

LIST OF TABLES vi 

LIST OF FIGURES viii 

CHAPTER 

I THE NEED FOR A NEW MATHEMATICS COUNSELING 
APPROACH TO SUPPORT COLLEGE STUDENTS 1 

II THE THEORETICAL CONTEXT FOR RELATIONAL 
MATHEMATICS COUNSELING 15 

III A NEW APPROACH: BRIEF RELATIONAL 
MATHEMATICS COUNSELING 60 

IV METHODOLOGY 103 

V AN ACCOUNT OF THE SUMMER 2000 PS YC/STAT 1 04 

CLASS 139 

VI UNCOVERING MATHEMATICS RELATIONAL PATTERNS: 
THREE PSYCH/STAT 104 COUNSELING CASE STUDIES 196 

VII DEVELOPING THEORY: STUDENT CATEGORIES AND 

WAYS OF COUNSELING 307 

VIII REFLECTIONS ON BRIEF RELATIONAL MATHEMATICS 
COUNSELING: EVALUATION OF THE PILOT STUDY AND 
RECOMMENDATIONS FOR FURTHER RESEARCH 341 

APPENDICES: 

A: Proposed Brief Mathematics Counseling Approach Chart 374 

B: Individual Assessment Instruments 375 

C: Class Assessment and Observation Instruments 388 

D: Informed Consent Forms etc 446 

E: Coding and Analysis 451 

F: ClassSeating 459 

G: PS YC/STAT 1 04 Instructor Handouts and Syllabus 462 

H: Class Data 471 

I: Class Calendar of Events 474 

J: Mathematics Counselor Tutoring Handouts 479 

K: Karen's Data 486 

L: Jamie's Data 491 

M: Mulder's Data 498 

REERENCES 503 



Ill 



ACKNOWLEDGEMENTS 

First I thank the instructor and the students of the statistics in psychology summer 
2000 class who so generously allowed me to pilot with them my new approach to 
mathematics support. While their names must remain anonymous I hope that this 
document reveals each one's unique learning history and approach to mathematics and 
the courageous changes each made to improve in that approach. 

My deepest gratitude and love go to my husband Robert, and daughters Christina, 
Andrea, and Margaret who have gone beyond all expectations in supporting my efforts 
and believing in my ability to complete this project. I could not have done it without them 
and can only guess at the impact of the sacrifices each has made to make it possible. 

A sincere thank you to my senior advisor and first reader Rebecca Corwin for her 
ongoing efforts in reading draft after draft of this manuscript often under considerable 
time pressure. Special thanks go to Cornelia Tiemey, my second reader, for her extra 
work in reading drafts and for her incisive critiques. Paul Crowley also deserves my 
thanks for his work as third reader, especially for helping me hone my counseling 
insights and skills. 

I am deeply gratefiil to Sarah Madsen Hardy and to Caroline Heller and the 
Lesley Ph.D. Fall 2003 writing group for their wonderfiil support in helping me fmd and 
develop my writing self. 

My thanks go to those with whom I work, especially to Margaret Pobywajlo and 
Roberta Kieronski but also to other colleagues, peer tutors, and students who have not 
only supported me in my long writing process but also have continued to teach me and 
deepen my insights into ways the learnmg process is embedded in our relational realities. 



IV 



Finally my deep thankfulness goes to Jesus, my extended family and friends, and 
my community of faith who have prayed for and supported me through this long process 
and have accepted me with the sometimes startling changes I have made. 

I would like to dedicate this manuscript to the memory of my parents Molly and 
Geoff Fraser. My mother inspired me with the vision to see potential in those who could 
not see it in themselves. My father challenged me to move beyond sentiment to expect 
and demand realistic effort and careful thinking of myself and of those with whom I 
work. 



Abstract 

Traditional approaches to college mathematics support focus on cognitive aspects 
of the student's approach and only incidentally address affective problems such as 
mathematics and testing anxiety. Because such affective conditions may be symptoms of 
underlying relational problems rooted in a student's learning history, I proposed a brief 
relational mathematics counseling approach that integrates cognitive constructivist 
mathematics tutoring into a brief relational conflict counseling framework (Mitchell, 
1988; Windschitl, 2002). 

I hypothesized that using this approach, professional tutors who also took on a 
role of mathematics counselor, could help underachieving students, during an 
introductory level mathematics college course, improve their approach to mathematics 
and avoid failure. This pilot study was conducted with an introductory statistics class at a 
small urban commuter university in the Northeast United States. Ten of the 13 students in 
the class volunteered for counseling; each participant had an average of 5 sessions. The 
counselor helped students explore their relationships with their mathematics selves, 
internalized presences, and interpersonal attachments while sessions focused on 
mathematics and course management. This new approach to mathematics support helped 
the counselor and the student become aware of mathematics relational patterns impeding 
the success of the student, and allowed both to develop constructive ways to change 
counterproductive behaviors. 



VI 



List of Tables 


Table 4.1 


Table 4.2 


Table 4.3 


Table 5.1 



Profile Summary of Students taking PSYC/STAT 104, 
Summer 2000 114 

PSYC/STAT 104, Summer 2000 Class and Research 
Schedule 116 

Conventions Used in Presentation of Transcripts 129 

Grades Throughout the Course of all Individuals in 
PSYC/STAT 104, Summer 2000 183 



Table 5.2 



Number of Individual Utterances during Lecture Portion 
of Classes and Final Grade 185 



Table 6.1 


Table 6.2 


Table 7.1 


Table 7.2 


Table 7.3 


Table Al 


Table El 


Table E2 


Table E3 


Table E4 



Focus Participants' Levels of Understanding of the 
Variable on the Algebra Test 300 

Focus Participants' Understanding of Arithmetic on the 
Arithmetic for Statistics assessment 301 

Criteria for Determining Level of Mathematical 
Preparedness of PSYC/STAT 104 Participants 308 

Emerging Categories of Mathematics Self Development 

310 

Criteria for determining Malleability of PSYC/STAT 104 

Participants 312 

Proposed Brief Relational Mathematics Counseling 
Summary 374 

Analysis scheme for Counseling Session Data: Student's 
Mathematical Relational Matrix 463 



Analysis of Lecture Session Student Exchanges with 
Instructor 464 

Analysis of Student's Problem Working Session 
Behaviors 465 

Protocol for Analysis of Exam Question Solutions 466 



Table HI 



Students' Expectations & Hopes in Relation to Effort, 
Grades and Scores, Summer 2000 461 



Vll 



Table H2 Student Tier (Tobias) and Category (Knowles) in 

Relation to Class Rank after Exam #1 and Pre- and Post- 
Statistical Reasoning Assessment (SUA) scores 462 

Table H3 Students' Pre and Post Positions on Feelings and Beliefs 

with Net Number of Changes 463 

Table Kl Karen's Individual PS YC/STAT 1 04, Summer 2000 

Participation Profile 494 

Table K2 Karen's Progress in Tests in Relation to Mathematics 

Counseling Interventions 495 

Table K3 Numerical Scores and Averages of Karen's JMK 

Mathematics Affect Scales Responses 498 

Table LI Jamie's Individual PS YC/STAT 104, Summer 2000 

Participation Profile 499 

Table L2 Jamie's Progress in Tests in Relation to Mathematics 

Counseling Interventions 500 

Table L3 Numerical Scores and Averages of Jamie's JMK Scale 

responses 505 

Table Ml Mulder's Individual PSYC/STAT 1 04, Summer 2000 

Participation Profile 506 

Table M2 Mulder's Progress in Tests in Relation to Mathematics 

Counseling Interventions 507 

Table M3 Numerical Scores and Averages of Mulder's JMK Affect 

Scale Responses 510 



VIU 



List of Figures 

Figure 5.1 Room and furniture configuration for PSYC/STAT104 

class, second floor, Riverside Center, Brookwood State 
University, Summer 2000 141 

Figure 5.2 Individual's scores on Test #1, with each student's score 

broken down into his Conceptual( multiple-choice Part I) 
score, out of 40; his Symbol score (on Part I) out of 8; 
and his Computational score (Part II) out of 52, total 
possible 100 172 

Figure 7. 1 Mathematics Self Development Categories of 

PSYC/ST AT 104 participants. 331 

Figure Fl Jillian's seating positions for the PSYC/STATS 1 04 

course, second floor. Riverside Center, Brookwood State 
University, Summer 2000 467 

Figure F2 The most usual seating choices of students for 

PSYC/STAT104 course, second floor. Riverside Center, 
Brookwood State University, Summer 200 468 

Figure KI Karen's pre and post scores on the pre Mathematics 

Feelings Survey in relation to class extreme scores. 496 

Figure K2 Karen's pre- and post-summary scores on the 

Mathematics Beliefs Survey in relation to class range 
scores. 496 

Figure K3 Karen's responses on the JMK Mathematics Affect Scales 

(in Mathematics Coimselmg Sessions 2 through 5) 497 

Figure LI Jamie's responses on Questions 1 1 and 12 on Exam #1 . 

Note her self-corrections in question 1 1 . 501 

Figure L2 Jamie's responses on Questions 13 and 14 on Exam #1. 

Note her self-corrections in questionl3. 501 

Figure L3 Jamie's solution to question 7 (d) Find the area of the 

figure. Algebra Test (Sokolowski ( 1997); see Appendix 
D). Note Jamie's uiitial error that she scratched out and 
replaced with the correct properly coordinated area 
solution. 502 



IX 



Figure L4 Jamie's mathematics pre- and post-feelings and beliefs 

Survey Profile Summary in relation to class range pre- 
and post-scores. 503 

Figure L5 Jamie's responses on the JMK Mathematics Affect 

Scales, Sessions 1-5 504 

Figure Ml Mulder's responses to the pre- and post-Mathematics 

Feelings and Mathematics Beliefs surveys in relation to 
class extreme scores. 508 

Figure M2 Mulder' s responses on the JMK Mathematics Affect 

Scales for Sessions 1,3, and 5. 509 



CHAPTER I 

THE NEED FOR A NEW MATHEMATICS COUNSELING APPROACH 
TO SUPPORT COLLEGE STUDENTS 

This research grew out of my experience as a mathematics learning specialist in 
academic support centers at two- and four-year colleges and universities in the Northeast 
United States. Over the years, certain students who came to me for help puzzled me. Some 
whose skills seemed inadequate or whose experience of mathematics seemed too damaging 
went on to succeed, while others succeeded at the introductory level but could not continue 
and eventually changed to a major that had fewer mathematics requirements or none at all. 
Too many of my students — ^typical students with no diagnosable learning disabilities — 
withdrew or failed. Again and again I noticed that academic proficiency alone could not 
explain my students' success or failure. I became convinced that there was another way of 
understanding how college students learn mathematics — one that my training and 
experience did not give me tools to address at that point. 

A young woman I will call Janet was one such puzzling student. A freshman who 
was taking a business precalculus course, Janet was in the practice of coming once or twice 
a week to the Mathematics Support Center during drop-in time to work on her homework. 
The peer tutors or I would check on her and sit down with her if she was struggling. One 
morning there were few students and no peer tutors present, just me, the professional tutor. 
Janet was sitting close to the table with her notebook and mathematics text on the table in 
front of her. As I went over to check on how she was doing, she pulled her hands out from 
under the table. Without thinking, I blurted out, "What were you doing?" Shamefaced, Janet 
replied, "I was counting on my fmgers." She was working on factoring a quadratic equation 
and was trying to work out what factors of 24 summed to II by tapping her fingers on the 



underside of the table. I asked why under the table — ^and not above, where she could see her 
fingers — and she told me that when she was six years old, in first grade at a parochial 
school, her teacher had rapped students' knuckles whenever she caught them using their 
fingers to help with arithmetic problems. The teacher had forbidden the use of fingers or 
other counting materials, taking the need for them to mean a student had not done her 
homework or memorized her addition facts. So Janet had learned to keep her counting 
hidden and had never committed her addition facts reliably to memory. 

Janet grasped quadratic equations — clearly she had the aptitude to memorize these 
arithmetic facts. The question was why she hadn't. At the age of 19, Janet was stuck in 
mathematical behavior that was now neither appropriate nor necessary, although it was 
sensible in the early grades. She was managing in her course, although it always seemed to 
be a quiet struggle and she never seemed confident of her outcomes. I began to wonder 
about how Janet's first grade teacher's treatment and her ongoing need to rely on hidden 
counting had affected Janet's view of herself as a mathematics learner. I wondered whether 
she was now aware that it is considered developmentally appropriate for first graders to 
count usmg physical objects. Or did she still believe, as she seemed to have then, that the 
teacher was justified in her knuckle rapping and that she was bad at mathematics, as the 
teacher implied? 

Janet was confident and doing well in her other courses; it was only in mathematics 
that she was struggling. Even within mathematics there seemed to be discrepancies in her 
confidence and achievement. She grasped difficult precalculus concepts such as the 
composition of fiinctions, but her tentative grasp of underlying arithmetic facts often seemed 
to undermine her confidence in her understanding of such advanced concepts. Despite her 



difficulties and unrealistic underconfidence, Janet did persist and earned a B~ in her 
course — not as high a grade as I felt she could have earned, however. Her persistence also 
puzzled me. I saw other low-confidence students, with no early trauma and with a sound 
grasp of the underlying arithmetic facts and concepts, who came intermittently to the 
mathematics support center, seemed to feel helpless to change their gloomy expected 
outcome, and ended up withdrawing or failing. 

Janet was the kind of student whom I often found myself wanting to help but not 
knowing how. I could recommend the upcoming mathematics anxiety workshop. Although 
anxiety did not seem to be at the core, Janet did have affective problems with mathematics 
that included anxiety. Likewise, I could help her master her precalculus content. But from 
past experience I knew that these interventions were unlikely to affect Janet's overall 
approach to herself learning mathematics. And yes, I could (and did) tell her about research 
and good practice in elementary education that showed that her teacher had been wrong and 
that Janet's use of fingers in first grade did not mean that she was bad at mathematics. But I 
suspected that simply communicating this information would not be enough to convince her 
that she had always been and was now able to understand and master mathematics. Her 
progress continued to be achieved at what appeared to be considerable emotional cost and 
little sense of personal ownership; she attributed her relative success to the peer tutors and 
me. 

The struggling students who inspired me to undertake the research described in 
this dissertation are in many ways typical of American college students. Regardless of 
their major, in most U.S. colleges or universities, students are required to take a 
mathematics course at an introductory college level for a liberal arts degree; some must 



go further for their major. More students withdraw jB-om or fail these courses than any 
other college courses (Dembner, 1996a, 1996b). Students who do not llilfill their 
college's mathematics requirements often abandon or change their academic and career 
goals. Students like Janet may persist in mathematics for their major, but fail to develop 
the confidence to apply it to related courses or in the workforce. 

Colleges and universities have attempted to address the problem of failing and 
withdrawing mathematics students and the alarming attrition rate' of students from 
mathematics and mathematics-related majors in college (Madison, 2001; National 
Research Council, 1991). The most prevalent assumption on the part of colleges and 
universities is that this failure and attrition can be attributed to students' deficient high 
school mathematics backgrounds. Perhaps they are taking classes for which they do not 
have the prerequisite knowledge. 

Increasingly since the mid-1980s, academic institutions have attempted to support 
struggling students and to encourage those who are more confident to continue studying 
mathematics by instituting placement testing; developing short courses to teach 
prerequisite mathematics and study skills; establishing learning resource centers that 
usually provide peer and professional tutoring for individuals and groups, instituting 
behavioral or cognitive counseling programs, and offering workshops focusing on study 
skills or testing anxiety (Boylan, 1999; Hadwin & Winne, 1996). Many also offer pre- 
college level developmental mathematics courses." All of these are efforts by the 
institution to reduce failure and enhance retention. Much of it may be seen to fall under 
the umbrella of what used to be called remedial but is currently called developmental 
education. '" 



From the perspective of the individual student, the outcomes of such efforts are 
uncertain, however. In my position as a mathematics specialist in the learning support 
center, a central piece of the university's failure reduction and retention effort, I see that 
while students who make strategic use of such resources can achieve at a higher level in 
mathematics coursework, many who need help are not strategic in accessing it. Furthermore, 
the predominantly cognitive, skills-based approaches that the learning support center offers 
are ineffective for understanding and addressing the problems faced by underconfident, 
anxious, or avoidant students. The measures currently in place fail to fiilly address the 
problems of college students like Janet — ^those with what I have come to understand as poor 
mathematics mental health" The number of students I encounter who need help of a kind 
not provided by current approaches leads me to believe that poor mathematics mental health 
may be central to our national failings in mathematics. 

The Learning Support Center Context 

The research that is described in this dissertation grows directly out of the 
questions that plagued me over the course of a 15-year career as a learning center 
mathematics specialist. My objective was to find new approaches to helping students 
struggling with mathematics in the specific context of a college or university learning 
support center. I believe that academic support center personnel are well positioned to 
apply a new, more holistic approach to helping students struggling with poor 
mathematics mental health because of the opportunities for professional tutors to work 
one-on-one with students, the separation from regular classroom dynamics, and the semi- 
autonomy of typical learning support centers that makes organizational changes and the 



piloting of innovative approaches easier. However, this study was designed with a lively 
awareness of the practical challenges of working with students in this setting. 

In academic support centers like the one where I work, mathematics learning 
specialists overwhelmingly focus on mathematical skills and concepts. The pressures of 
everyday practice in an academic support center, the urgency the students feel because of 
the limited time available, and the importance of mathematics as an academic gatekeeper 
combine to create among academic support personnel a tendency toward unreflective 
pragmatism (Lundell & Collins, 1999). This pragmatism is characterized by only 
incidental assessment of affective issues as well as limited mathematics assessments. This 
short-term view leads to a default tendency to focus only on the course mathematics, 
especially on procedures and skills rather than understanding. Mathematics tutors are 
under great time pressure: Although students may make ongoing weekly appointments, 
we see the typical student only when he" chooses to come in. The incidental nature of our 
contact with the student exacerbates the problem. For too many students, this approach is 
not working adequately. 

The problem is not that mathematics learning specialists do not know about 
cognitive and affective factors significant for achievement, but rather that we know them 
abstractly and as separate factors, and lack an approach for gaining, prioritizing, and 
using this knowledge effectively. In addition, 1 increasingly had the troublesome sense 
that the cognitive and affective expressions that we see (e.g., Janet's fmger counting and 
underconfidence) may be symptoms rather than causes of a student's real difficulties. We 
deal daily with the interaction of these overt and hidden factors and their meaning for a 
particular student, but this meaning often eludes us. The quick diagnosis of a student's 



central problem, whether overtly cognitive or affective, underlying, or an interaction of 
these, is a particular challenge in the learning center context. 

For all of these reasons, it is unusual in mathematics academic support to find out 
about students' mathematics learning histories and the understanding, beliefs, attitudes, and 
habits that they developed as a result. My frustration with my limited ability to help Janet 
and others like her grew at the same time as I began to recognize clues to the puzzle. More 
and more, I became certain that the cognitive approach predominant in my field was not 
enough to understand how and why college students undertake to learn mathematics. How 
could I address the root source of Janet's arithmetic problem, her approach to coping with 
the problem over many years, and her evidently low mathematics self-esteem? I began to 
see that I needed a fiiller understanding of Janet's history and its effects on her present 
mathematics experience in order to understand what I, as a mathematics learning specialist, 
could do to help her change her mind about herself as a mathematics learner. 

I began to wonder if affective issues and learning histories might be important 
determiners of achievement in mathematics among typical college students. I suspected that 
cognitive outcomes were related not only to academic preparation, but to relational 
dynamics and affective experiences: an elementary school teacher who humiliated a student 
for asking a question or a parent who told a student that she inherited the family "we-cannot- 
do-math gene." Experiences like these may lead to otherwise inexplicable gaps in basic 
number facts and number sense or hazy understanding of the algebraic variable. Why did 
certain students fmd themselves unable to think, interact, or connect with the instructor? 
Could it be that students with otherwise adequate mathematics skills and aptitudes are 
limited by unconscious forces Imked to earlier mathematics learning experiences that cause 



8 

them to repeat counterproductive practices? Might one defining negative experience with a 
teacher in grade school or in high school affect a student's lifelong learning of mathematics? 
How might poor preparation interact with a student's mathematics identity to affect his 
approach in the current course? 

I wanted to fmd out what would happen if a mathematics learning specialist did 
have the opportunity to delve into these questions in the learning center context. What 
would be the result on a student's mathematics achievement in the current and future 
semesters if I were able to offer support based on a more holistic picture of that student as 
a mathematics learner? If mathematics learning specialists could fmd ways to understand 
and help the whole person rather than dealing with his parts — ways to address the 
mathematics mental health of their students — ^we might be able resolve these problems 
and more effectively and reUably help him go on to achieve long-term mathematics goals. 

The Study 

In my capacity both as a mathematics learning specialist and a doctoral student, I 
have searched for ways to understand students' mathematics mental health, diagnose their 
difficulties, and help them holistically and effectively. Through the study that is described in 
this dissertation, I have sought to create and test a more holistic approach of academic 
support that would help the many students I encountered with academic mathematics 
problems rooted deeply in relational conflict and other traumas that thwarted the 
development of their mathematical identities. 

This research was based on the hypothesis that an adequate knowledge of the student 
as a whole person doing mathematics may be a pivotal part of academic support personnel's 
plan for understanding and supporting him through his mathematics course. This hypothesis 



led to four research questions that are mformed by the set of challenges particular to an 
academic support center setting: 

1 . What does a mathematics learning specialist need to know about a student in 
order to understand him as a whole person doing mathematics? 

2. What processes can be used to gain this understanding quickly while he is taking 
a mathematics course? 

3. How can a mathematics learning specialist use this fialler understanding of the 
student to help him in the mathematics course he is taking? 

4. What does a mathematics learning specialist need to understand about himself as 
a counselor and tutor in order to help the student succeed? 

The search to answer these questions and thus understand and effectively intervene 
in each student's complex interactions between his mathematics affect and cognition led me 
outside the narrow boundaries of the field of mathematics academic support. The field of 
counseling psychology — in particular, relational psychotherapy — emerged as providing the 
most perceptive ways to understand the effects of students' mathematics learning histories 
on their current learning challenges. In chapter 2, 1 discuss the work of scholars I have 
drawn from. By adapting theories and practices of relational psychotherapy to mathematics 
learning, and then combining these new methods with the cognitive approaches that I had 
been practicing for years, I arrived at a brief relational mathematics counseling approach. I 
describe the development of this approach in chapter 3. To investigate the approach, I 
piloted it with students taking a summer introductory- level statistics course taught at a small, 
urban, commuter state university in the Northeast. The remainder of the dissertation 
describes and discusses the study itself — ^the use of case study methodology and the criteria I 



10 

used in my choice of particular cases to present in chapter 4; the presentation of the class as 
the case that creates the context for the individual cases in chapter 5; the individual cases in 
chapter 6; analysis of results and developing theory in chapter 7; and evaluation, limitations 
and implications of the study, as well as recommendations for further research in chapter 8. 

The goal of this study was to develop, pilot, and evaluate a mathematics 
counseling approach based on brief relational therapy approaches (with cognitive therapy 
and developmental psychology contributions) designed to help individuals attain sound 
mathematics mental health and success in reaching their own mathematics goals. This 
involved identifying, adapting, and developing instruments and approaches that explore 
students' mathematics learning; their history, beliefs, and attitudes about learning; and 
their relational patterns as they participate in an introductory level college mathematics 
course. Students engaged in a brief course of mathematics relational counseling with me 
as the mathematics counselor using these instruments and approaches. 

This study contributes an approach to the field of mathematics academic support 
that combines aspects of mathematics and personal therapy approaches drawn from 
cognitive, affective, and relational theory. It is designed to help college academic support 
staff understand and help the student as a whole person doing mathematics. It combines 
what are typically considered to be quite unrelated, disparate elements of mathematics 
learners and those who help them, that is, mathematics cognition and affect expressed in 
distinctive relational patterns (his, mine, and ours). The results provide some prelimmary 
data to establish groundwork for the development and use of this individual counseling 
approach to improve students' mathematics mental health and success in required college 
mathematics courses. 



11 

My goals can be further summarized thus: 

1 . To identify, adapt, and develop instruments and approaches that explore students' 
mathematics learning, their history, feelings, attitudes, and beliefs about learning, 
and their relational patterns as they may affect progress in an introductory-level 
college mathematics course; 

2. To pilot a mathematics counseling approach based on brief relational therapy 
approaches (with cognitive therapy and developmental psychology contributions) 
with the goal of helping individuals attain good-enough" mathematics mental 
health and success; and 

3. To evaluate assessment and treatment instruments and approaches, and more 
importantly, the brief relational mathematics counseling approach itself 

Over the past few years, my colleagues have looked at me quizzically when 1 tell 
them that my research explores how relational therapy that is rooted in Freud can help 
college students achieve in mathematics. Admittedly, my approach is quite 
unconventional. On the surface, the teaching and learning of mathematics seem to have 
little to do with the murky realm of unconscious motivations and relational conflicts. But 
when 1 observed my students' behavior, addressed their achievement problems as 
symptoms, and asked them to talk to me about how they felt about their teachers, their 
peers, themselves, and the subject of mathematics itself, the results were rife with 
conscious and unconscious motivations that were often in conflict, and counterproductive 
relational patterns in which students seemed stuck. 

While there are many tools to assess how affect effects achievement in 
mathematics and cognitive and behavioral treatments to address problems, to my 



12 

knowledge the only practitioner who has attempted to understand how mathematics 
issues can be addressed using a holistic individual approach based in Freudian 
psychotherapy is Lusiane Weyl-Kailey (1985), a Parisian psychotherapist who had been a 
mathematics teacher. Her work was conducted in a clinical setting with school children 
whose psychological and emotional disturbances she found to be connected with their 
mathematics learning problems. She used psychopedagogy — an integration of Freudian 
therapeutic and pedagogical approaches — to understand the psychological effects of 
mathematics on her clients in order to improve both their mathematics learning and their 
psychological health (Tahta, 1993; Weyl-Kailey, 1985). While she is a psychotherapist 
who brings her understanding of mathematics pedagogy into her therapy with disturbed 
students who had mathematics learning issues, I am a mathematics educator who 
proposes to bring Freudian-related relational conflict therapy as a new approach into the 
learning support of average mathematics students who have affective and relational 
barriers to their mathematics learning. 

In this study I show that close psychological attention to unconscious motivations 
and conflicts is applicable not only for those whose mathematics learning problems may 
be related to personal emotional disturbances but more generally for ordinary college 
students whose psychological functioning is within the range of "normal," and this 
counseling approach may be appropriately delivered in the educational setting. In the 
following pages, I will define a mathematics selfihaX we all have, no matter how deeply 
neglected, damaged, or denied. I will explain how a teacher or tutor can be like a parent 
in the psychological development of this mathematical self It is my hope that the theories 



13 

I have explored and the approach I have piloted will open the door to a new way of 
thinking about academic support that nurtures and heals students' mathematics selves. 

When I begin to describe my work and my dissertation project, many people 
(university colleagues, students, friends, acquaintances, fellow partygoers or fellow 
church members) want to tell me their mathematics story. Each one wished that when 
they were struggling with the mathematics course that ended or changed their career 
aspirations, they had had someone knowledgeable in mathematics, mathematics 
pedagogical research fmdings, and relational counseling approaches who had been able to 
help them understand and get over their fears and low confidence so that they could 
proceed with their mathematics learning. For others the topic is so painful that they have 
to change the subject or walk away. And there are some who have a story of struggle and 
triumph and a few who never or rarely struggled, almost always "getting it" and 
succeeding. It is for the many who, for want of someone who could listen knowingly and 
intervene strategically, performed poorly or avoided or failed in the mathematics they 
needed, that I pursued this dissertation research. 



14 



' Of the 3.6 million U.S. mathematics students in ninth grade in 1972, only 294,000 persisted to al-Ievel 
mathematics courses as freshmen in college in 1976. Only 1 1,000 continued to graduate with a bachelor's 
degree in mathematical sciences in 1980, and 2,700 succeeded in graduating with a master's degree in 
1982. See National Research Council, 1991, p. 19, Figure 5. These figures are relatively dated but the 
current progression appears to be similar. 

" Currently, abnost all community colleges and more than 60 percent of other colleges and universities in 
the United States offer developmental courses in mathematics, writing, study skills, and in some cases 
reading (Bibb, 1999;Dembner, 1996), mathematics developmental courses being the ones most enrolled in 
by freshmen, however (Phipps, 1998; Madison, 1990). 

"' In recent discussion of the evolution of developmental education in colleges and universities, Payne and 
Lyman note that the preference for the term "developmental" over "remedial and developmental" was 
formalized in 1976 when the name of the professional journal was changed to reflect that. They point out, 
however, that the field has been known by many other names in its long history (Payne, 1996). Higbee 
(1996) sees the essential difference between "remedial" and "developmental" as the difference between "to 
correct a previous wrong" and "to promote the growth of students to their highest potential" (p. 63), that is, 
the difference between a deficit and a growth orientation. 

™ Sheila Tobias (1993) uses the term "math mental health" to refer to a person's "willingness to learn the 
mathematics [he] needs when [he] needs it" (p. 12), using it as the criteria to assess a student's mathematics 
ftmctioning beyond the cognitive, fri contrast, in adopting her term I include under it all aspects of a 
student's mathematics ftmctioning including cognitive factors. 

" In odd numbered chapters I use the masculine, "he," "him," and "his" for the third person singular generic 
pronoun. In even numbered chapters I use the feminine, "she," "her," and "hers." 

" I have adapted the use of Winnicott's (1965) term "good-enough" for this study. A fiill discussion of his 
use of it and my adaptation comes in chapter 2. 



15 

CHAPTER II 
THE THEORETICAL CONTEXT FOR A RELATIONAL COUNSELING APPROACH 

I identified in chapter 1 the central problem that learning specialists face when we 
try to help students achieve their potential in college-level mathematics. We focus 
narrowly on course-related mathematics skills and concepts; we may help the student 
improve her grade but fail to understand and help her' as a whole person doing 
mathematics. The focus is so much on helping her pass her course that we do not stop 
long enough to Usten and understand what is really preventing the success she aspires to. 
What if I had the opportunity to hear her story and understand how certain people or 
experiences might have affected how she is doing mathematics now? What if I knew how 
to help her unravel herself from beliefs and behaviors that seemed to be standing in the 
way of her success, beliefs and behaviors that had developed over the years as the result 
of those people and experiences? I determined that if there were a way to use an 
individual counseling approach that could be incorporated into regular mathematics 
support offered through the learning support center, the problem I had identified might be 
resolved. 

I was then faced with the task of finding and/or developing a counseling approach 
or approaches adaptable to the central mathematics learning task, compatible with the 
educational setting, and, most importantly, perceptive of underlying causes. In this 
chapter, I describe my search for such a counseling approach and demonstrate how my 
research into existing theories in the fields of education and counseling psychology 
provided the insight I needed to help the whole person doing mathematics. 



16 

CONTRIBUTIONS FROM THE FIELD OF 
MATHEMATICS EDUCATION 

First I asked if researchers and practitioners in the field of mathematics education 
had also perceived the problem I had identified and, if so, what they had done about it. I 
found that there is a large body of research on cognitive (cf Hiebert & Lefevre, 1986; 
Piaget, 1969) and affective (cf McLeod, 1989, 1992, 1997; McLeod & Ortega, 1993) 
factors of mathematics functioning and on the relationship between cognition and affect 
(cf Boaler, 1997; Buxton, 1991; Skemp, 1987). Pragmatic approaches to improving 
students' mathematics functioning problems range from those that focus primarily on 
cognitive problems (changing mathematics pedagogy or curricula), through those that 
focus primarily or affective problems (chiefly alleviating emotional symptoms such as 
anxiety), to those that focus simultaneously on both cognitive and affective problems 
(some dealing with affect and cognition separately, cf Nolting, 1990), others dealing 
with them as interconnected factors (cf Carter & Yackel, 1989; Tobias, 1993). 

Researchers and practitioners of mathematics education concur that a student's 
mathematics functioning involves both cognitive and affective factors, although there is 
little clarity on how these factors interact (cf McLeod, 1992; Schoenfeld, 1992). As a 
minimum, they suggest m order to understand how a student is functioning 
mathematically, a mathematics learning specialist needs to know what the student 
understands of the prerequisite mathematics, how well she can apply that background 
understanding in learning new mathematics concepts and procedures, and any affective 
orientations she has developed that might affect that learning process. 



17 

Cognitive Factors 

To know what the student understands of the prerequisite mathematics, a college 
usually attempts to gauge her current level of competence using high school records and 
course-taking history, a college-devised placement test, Scholastic Aptitude Test (SAT) 
or American College Test (ACT) mathematics score, an interview, or some combination 
of these. If course placement is mandated by this process, the student and the 
mathematics learning specialist have some assurance that the level of difficulty of the 
current course is within range of her capabilities. Other aspects of the student's cognitive 
processing known to have affected her mathematics learning and present achievement 
such as her preferred mathematics learning style," concept developmental levels, and 
long- and short-term memory are generally not assessed, so Uttle is knovra except what a 
learning specialist observes in tutoring. A student's awareness of her own learning 
processes and her strategic study skills, when developed in relation to current 
coursework, have also been found to be significant cognitive factors that are often linked 
with achievement (Hadwin & Wiime, 1996). 

The cognitive effects of the mathematics teaching approaches the student has 
experienced may be even more significant. Students who have experienced 
predominantly procedural rather than conceptual'" teaching approaches are likely to see 
mathematics learning as memorization of procedures rather than understanding of 
concepts and their connections, making security in the mathematics they know tenuous 
and new learning more difficult (Boaler, 1997; Skemp, 1987). Students who have 
experienced a teacher transmission and textbook exercise approach rather than a student- 



18 

centered, problem-solving approach are not likely to have developed effective strategies 
for approaching new mathematics learning (cf. Schoenfeld, 1985, 1992). 

While it seemed that in my approach I would need to be mindful of all these 
cognitive fectors as potentially significant in a student's success, I was concerned about 
the challenge of identifying aspects of mathematics affect that might be just as significant 
and understanding how these factors interacted. 

Mathematics Functioning: Affective Factors are Crucial 

In academic support, I had found that research on affect — beliefs, attitudes, and 
feelings — and its effects on students' mathematics learning and achievement is even 
more difficult than research on cognition to translate into understanding an individual's 
beliefs and feelings about her mathematics learning. It also seems more difficult to apply 
this understanding to developing a plan to help her succeed in her course. Mathematics 
and testing anxiety, locus of control, issues of learned helplessness, attribution, as well as 
achievement motivation are all affective factors that have been demonstrated to be factors 
in mathematics achievement (Dweck, 1975, 1986; Hembree, 1990; X. Ma, 1999). 
Mathematics cognitive psychologists like Skemp (1987) who look at students from the 
perspective of mathematics cognition have identified negative affective orientations and 
outcomes linked with teaching and learning approaches. Others like Buxton (1991) who 
have looked at students from the perspective of thefr affective difficulties with 
mathematics have identified problems in their cognition and cognitive learning 
approaches. To make this even more complex, demographic characteristics (gender, 
socioeconomic status, age, first language, and race or ethnicity) may interact to magnify 



19 

or minimize individual effects of a student's past experience on her achievement 
(Secada, 1992). 

Existing Approaches that Attend to both 
Affect and Cognition 

Mathematics support personnel and researchers have struggled to understand 

interactions among students' cognition and affect on their mathematics resilience and 

achievement. Much work has been done in the attempt to develop ways of helping. From 

this research, four major approaches have emerged. Each is a pragmatic attempt to help 

adults overcome their underachievement, aversion, and fear of mathematics. The 

approaches yield important information for my work, although I found that their 

usefulness is limited by the fact that they are either not directly applicable to the setting 1 

am investigating or they do not provide an adequate framework for holistic understanding 

and counseling. 

The First Approach: Freestanding Anxiety Reduction 
Workshops or Short Courses 

The most typical approach is a freestanding" course or workshop where the 
participants do mathematics as they tell their mathematics stories. Through this they 
become conscious of their own affect, habitual reactions, and beliefs about mathematics 
and the effects on their mathematics identities (cf Buxton, 1991; Kogehnan & Warren, 
1978; Tobias, 1993). 

To this list Carter and Yackel (1989) added another: adults' enculturation in and 
orientation to mathematics learning. They used Skemp's (1987) categories, distinguishing 
between an "instrumental"^ mathematics orientation (characterized by a "just teach me 
how to do it — I don't want to understand it" procedural approach) and a "relational""' 



20 

mathematics orientation (characterized by an "I want to understand why it is so and how 
this relates to what I already know" conceptual approach). They found that an 
instrumental approach was generally linked to heightened anxiety and passive behaviors, 
while students taking a relational approach used active problem-solving strategies and 
make positive attempts to construct mathematical understandings. Carter and Yackel used 
journal writing, cognitive constructivist problem-solving approaches, and cognitive 
behavior therapy techniques such as cognitive restructuring to help participants move 
from an instrumental (procedural) to relational (conceptual) orientation to mathematics. 
They found that students who made this change also experienced a significant reduction 
in mathematics anxiety."' 

One important limitation of Tobias's or Carter and Yackel' s approach for my 
work, however, is the fact that it is freestanding and thus not linked with a college course. 
Ahhough participants tend to become less anxious and gain confidence, there may be 
little positive effect on their achievement in a college courses taken concurrently (E. 
Yackel, personal communication, January 21, 2000). Notwithstanding, these researchers 
do contribute some significant elements to the design of my approach: Practitioners like 
Tobias or Carter and Yackel stress group work and focus on identifying (and challenging) 
counterproductive thoughts and behaviors at the conscious level. Their successful use of 
cognitive counseling techniques such as hypothesis testing of faulty beliefs"" and 
cognitive restructuring" prompted me to investigate cognitive counseling further for its 
possible contributions to my approach. Most importantly this process integrates focus on 
research-supported conceptual mathematics approaches with linked affective outcomes 
rather than treating cognition and affect separately. I determined to investigate how I 



21 

might incorporate Carter and Yackel's successful use of constructivist problem-solving 

mathematics pedagogy to change counterproductive mathematics orientation and affect 

into my approach in a college learning assistance context. 

Second Approach: Study Skill and Anxiety Reduction Co-Courses 
Linked to a College Mathematics Course 

The second approach noted in the literature consists of a second course or lab 
linked to a college mathematics course. Addressing counterproductive beliefs and habits, 
these co-courses focus on developing skills for mastering the mathematics content of the 
college course and are typically effective m improving students' achievement (cf 
Stratton, 1996; Nolting, 1990). Even further improvement in achievement resulted for 
students diagnosed with high external locus of control" when a brief course of individual 
cognitive counseling aimed at internalizing locus of control and reducing helplessness in 
the mathematics learning situation was provided (Nolting, 1990). This success 
encouraged me in my pursuit of an individual counseling approach, but my experience 
told me that it is not only students with high external locus of control who could benefit 
from individual counseling; students like Janet (see chapter 1) have other forms of 
emotional impediments to achieving their mathematics potential. 

1 noted the consistent findings of Stratton, Nolting, and other researchers (Hadwin 
& Winne, 1996) that students benefit more from study skills and negative affect reduction 
courses or workshops that are linked to a particular academic course they are taking 
simultaneously than from freestanding offerings that are not specific to a particular 
course. This finding was a key incentive for me in pursuing an approach that could be 
tailored to a particular course the student was taking and delivered simultaneously. 



22 

A Third Approach: Mathematics Instructors Addressing 
Affect in the Classroom 

The third approach involves the mathematics instructor herself incorporating 
mathematics joumaling and/or history takmg, open discussion of feelings, and conceptual 
understanding and problem-solving development into the course curriculum (cf 
Rosamond, cited in Tobias, 1993, pp. 232-236). This type of self-contained situation 
where instruction and support to overcome affective and cognitive challenges are 
combined in the classroom is unusual. Its feasibility depends on the availability of 
instructors who understand not only the importance of affect in mathematics learning but 
also how to incorporate such understanding into classroom instruction of adults while 
also covering the material mandated by the college mathematics department. I speculated 
that if I were supporting students of such an instructor in the learning support center, I 
might still find individuals for whom the whole class treatment of affect was not 
sufficient. Importantly however, helping these students individually access and address 
the core of their difficulties would almost certainly be facilitated by the significance 
placed on affective issues by the instructor. This understanding made me conscious of the 
importance of attending in my design to the effect of the current instructor and classroom 
approaches on a student I was helping. 

Approach Four: Individual Counseling Approaches 

The fourth approach focuses on work with individuals who experience 
psychological disturbances triggered by mathematics learning or directly impacting their 
mathematics learning. There has been a long tradition of the use of behavior and cognitive 
behavior counseling with individuals and groups adversely affected by mathematics anxiety. 



23 

using such techniques as desensitization, guided imagery, and relaxation training (Nolting, 
1990; Richardson & Suinn, 1972). 

Integrating Attention to Affect with Attention to Cognition: Summarizing Mathematics 
Practitioners' Contributions — Implications for my Approach 

There are particular mathematics education researchers and practitioners who 
have studied concepts important to me as I developed this approach to college students' 
mathematics mental health. Some were pivotal. Especially important is Carter and 
Yackel's (1989) and Tobias's (1993) finding that participants must engage and succeed in 
conceptual mathematics in order to improve their view of themselves as mathematics 
learners and their mathematics mental heakh. 

Factors related to mathematics fimctioning are expressed differently and lead to 
different outcomes in different learning contexts. The helplessness that Nolting (1990) 
noted in students with external locus of control as well as the passivity that Carter and 
Yackel (1989) observed in instrumental (procedural) mathematics learners have been 
linked with performance motivation"' in achievement situations (Dweck, 1986). 
However, I noted that these results must be sensitively interpreted, since helplessness has 
also been linked with learning motivation in high-achieving girls subjected to over- 
procedural teaching. Understanding these factors within the current mathematics 
classroom context and Ustening to the student helps avoid thoughtless direct application 
of large group experimental findings to the individual (cf Boaler, 1 997). 

The need for sensitivity again emphasizes the need for a whole-person approach 
that may be conceptualized in terms of a students' mathematics mental health (see 
chapter 1, endnote iv). Whether the designers of these approaches whom I have cited 
state it explicitly as a goal or not, their workshops and co-courses that included in-class 



24 

journaling and discussions of emotion and in some cases narrowly specific individual 
cognitive or behavioral counseling, helped participants to varying extents to become 
aware of their mathematics mental health challenges. To the extent that participants were 
supported in addressing these challenges, their mathematics mental health often 
improved. Some even became willmg (and able) to learn the mathematics they needed 
when they needed it (Tobias, 1993). In none of the approaches examined here, however, 
was there the opportunity for an individual to explore her unique mathematics mental 
health challenges with a suitably qualified professional while she was engaged in a 
college course. 

My focus on mathematics mental health as a way of conceptualizing students' 
overall mathematics ftinctioning had become clear from examination of these approaches, 
and affirmed for me the need in the field for an individual approach to helping a student 
while she was taking a course. A logical next step was to identify or develop an 
individual counseling approach that could provide a fi-amework for simultaneously 
providing the mathematics cognitive support and acknowledging and addressing 
students' affective problems. 

CONTRIBUTIONS DRAWN FROM COGNITIVE THERAPY (CT) 

When I explored the wide range of counseling psychologies that might be 
applicable, the approach that first drew my attention was cognitive therapy (CT), which 
was developed by Aaron T. Beck in the 1970s. I knew of CT's links with cognitive 
psychology and had already noted the use of a number of its techniques in alleviating 
mathematical affective problems. I found that techniques of cognitive behavior therapy 
(CBT) which developed from Bandura's (1986) use of social-cognitive theory to merge 



25 

behavior therapy and cognitive therapy, had also been used effectively in educational 

settings. I resolved to explore how CT theory (and CBT, where applicable) and 

techniques might be adapted for my use as a basis for mathematics counseling. 

Exploring Cognitive Therapy (CT) Theory as a 
Framework for Mathematics Counseling 

I wanted to help students become aware of how their past experiences and current 
beliefs about themselves might be affecting their mathematics functioning; I also sought 
ways to help students modify the underlymg orientation and overt behaviors that were 
preventing their success. I wondered if cognitive therapy (CT) and cognitive behavioral 
therapy (CBT) approaches with their focus on helping clients change their 
counterproductive ways of thinking and behaving might offer what I was looking for. 

CT conceives of awareness as a continuum rather than a dichotomy separating 
conscious from unconscious experience. Beck (1976), the founder of CT, proposed that 
"Man has the key to understanding and solving his psychological disturbance within the 
scope of his own awareness" (p.3). Beck argued that his CT approach would change the 
person's view of herself from "a helpless creature of [her] own biochemical reactions, or 
of blmd impulses, or of automatic reflexes [as he contended that Freudian theorists 
claimed]" to a person "capable of unlearning or correcting" the "erroneous, self-defeating 
notions" she had previously learned such as, in this context, her supposedly genetic 
inability to do mathematics (p.4). CT focuses more on how the patient distorts reality than 
on why. In therapy, "the therapist helps a patient to unravel his distortions m thinking and 
to learn alternative, more realistic ways to formulate his experiences" (p. 3). 



26 

Cognitive Therapy (CT) and Mathematics Depression 
Students' emotional difficulties with mathematics often seemed to me to be 
different from traditionally recognized mathematics anxieties or phobias. When I 
examined CT's conceptualization of depression, I realized that much of what I had 
observed could be seen as a type of situational mathematics depression. I have seen in 
students' expressed negative views of their mathematics selves, mathematics worlds, and 
mathematics futures, a more local or situational counterpart of negative views of one's 
self, one's world, and one's future that, according to Beck (1977), characterize a 
depressed person's orientation to life. I had also noticed that (as Beck, 1977, and 
Seligman,1975, did in clients with generahzed depression) this mathematics depression 
was almost invariably linked with helpless beliefs and behaviors m the mathematics 
context. The promise of being able to differentiate depression from anxieties in the 
mathematics learning setting added an important piece to my approach. 
Other Cognitive Therapy (CT) Contributions 
Dweck (1986), Beck (1977), and others have emphasized the importance of and 
techniques for identifying and verbalizing erroneous and negative automatic thoughts in 
order to test their veracity and defuse their power. Nolting (1990), Buxton (1991), and 
others suggest the importance of students becoming consciously aware of their own 
affect. Buxton (1991), Tobias (1993), Carter and Yackel (1989) and Stratton ( 1 996) 
observe the therapeutic value of recognizing one's already existing mathematics aptitude 
and finding oneself capable of doing mathematics. Cognitive and cognitive behavior 
therapy (CT and CBT) and counseling techniques have been used effectively and 
extensively in educational settings. As noted above, Tobias (1993), Carter and Yackel 



27 

(1989), and others use CBT techniques such as cognitive restructuring, hypothesis testing 

of faulty beliefs, assigning aflbctive homework, and desensitization in their mathematics 

anxiety reduction workshops. Nolting (1990) also demonstrates the efficacy of a limited 

CBT cognitive restructuring approach (see endnote ix) to reduce the external locus of 

control of certain beginning algebra students. 

I determined that each of these CBT techniques might become part of my toolbox 

to help students. CT in theory (though not always in practice) takes a constructivist, 

problem-centered approach in that the client is seen to be the author of her own cure and 

the counselor becomes a coach as they collaboratively identify key problems that the 

client works to solve. This is the stance I chose as a mathematics learning specialist, to 

take with my students. I saw an important advantage of CT/CBT's brief therapy mode in 

college mathematics counseling. A course of CT/CBT therapy ranges from as few as 

three to as many as thirty sessions, but is typically conducted in ten to twenty sessions, a 

promising match for a college semester timeframe. 

Limitations of Cognitive Therapy (CT) as a Framework for 
Addressing Mathematics Mental Health Issues 

CT still left unaddressed, however, how a student's present patterns of 

mathematics fianctioning may have been influenced by her past experiences, which I had 

identified as crucial for understanding and helping mathematics students. CT does not 

consider the present role of the unconscious in sabotaging conscious motivations. I have 

found that students are not dealing only with erroneous automatic thoughts that can be 

identified and reasoned with; they often seem influenced by unconscious motivations out 

of their awareness that stem from their past experiences and that are in conflict with their 

conscious desires. As a mathematics tutor relating with the student, I also find myself 



28 

reacting and behaving in ways that puzzle me. In CT I did not find a way of 
understanding these aspects of the student or myself or our interaction. 

CT and CBT theorists contend that understanding the origins of a psychological 
problem is not essential for producing behavior change (Wilson, 1995). The CT approach 
thus helps identify and deal with symptoms but does not provide a way to unearth the 
root of the problem. Perhaps, though, I reasoned, more than behavior change might be 
needed for a student to succeed in mathematics. When I am confronted with a student's 
puzzling behaviors she may be unaware of and contradictory automatic thoughts that she 
does not even understand, it may be difficult to find ways to refute them even with good 
present evidence or research or logic. With some, resolving the puzzle may require an 
understanding of its begiimings and its developmental history. 

CT/CBT counselors do not see a need to investigate unconscious motivations and 

internalized relationships, nor do they examine present relationships to find clues to the 

person's difficulties. It is precisely these motivations and relationships that I 

hypothesized were key contributors to understanding a student's mathematics mental 

health challenges. Although CT/CBT provided invaluable elements, I concluded that CT 

could not supply the overarching framework for a holistic appraisal of a student's 

mathematics mental health. 

CONTRIBUTIONS FROM RELATIONAL CONFLICT 
PSYCHOANALYTIC THEORY 

In recognizing the need to address root causes of mathematics affective problems, I 

returned to theorists of mathematics affect such as McLeod (1992) and looked more closely 

this time at their endorsement of classical Freudian-type analysis and counseling approaches 

albeit for cases of extreme mathematics emotionality (see McLeod, 1992, citing Tahta, 



29 

1993). In cases of severe disturbance some mathematics educators and therapists have 
looked at or advocate looking at the role of individual students' unconscious in their 
mathematics learning difficulties (cf Buxton, 1991; McLeod, 1992, 1997). As 1 noted in 
chapter 1, Weyl-Kailey (1985) uses Freudian psychoanalytic techniques in a clinical setting 
to probe and remediate puzzling mathematical behaviors as she uncovers and treats related 
psychological disturbances."" Weyl-Kailey and others (see endnote xii) found that attention 
to students' unconscious motivations gives insights that other approaches do not. These 
researchers did not, however, use such approaches to understand and help "normal," 
struggling college students in the educational setting succeed in their current course, and it is 
these "normal" students I planned to help. 

Because my interest was in the mathematics mental health of ordinary students, not 
just those with extreme difficulties, 1 had earlier rejected the utility of psychoanalytic theory. 
I found no critical tradition in mathematics education that understood mathematics affective 
and cognitive problems as symptoms of underlying causes rooted in each student's learning 
history and expressed in her current patterns of behavior and relationships. But I now saw 
the promise of psychoanalysis in its attention to the unconscious and the present effects of 
the past on everyone. Indeed McLeod (1997) noted with interest Buxton's (1991) suggestion 
that some struggles of such ordinary students with mathematics might well be understood in 
terms of Freud's concept of the superego. I resolved to explore Freud's theory and the 
theories that evolved from it. 

The work of Stephen A. Mitchell (1988) emerged as highly relevant to my 
research because it used a form of relational conflict psychotherapy derived from 
Freudian psychoanalysis to help ordinary adults who had goals but were so embedded in 



30 

relational patterns with themselves and their significant others (both mtemal and 
external) that those goals were not being fulfilled. Rather than seeing people through a 
classical Freudian lens as largely driven by mstinctual pleasure-seeking and aggression 
drives that continually engender internal conflict along a largely predetermined 
developmental path, Mitchell's (2000) relational conflict theory recognizes that people 
are hardwired for human relationships and that their drives, motivations, and conflicts are 
focused around developing and maintaining those relationships with others and with 
themselves. In 1988 Mitchell integrated the three major relational strands of 
psychotherapy that emerged from Freud's classical psychoanalysis: self psychology, 
object relations, and interpersonal psychology. Each of these strands emphasized one 
dimension of what Mitchell termed as a person's relationality or her current behavior that 
are the outcome of the development of her self her external and internalized objects, and 
her interpersonal attachments (Mitchell, 2000). When I considered these dimensions in 
the context of a student's mathematics learning experience, I interpreted them as follows: 

1. Mathematics self or selves; 

2. Internalized mathematics presences or objects; ™^ and 

3. Interpersonal mathematics relational or attachment patterns. 
Understanding a student's mathematics relational dimensions, how they are 

positioned in relation to each other, and how they interact with one another to express her 
relationality might provide the insight into the origin and development of her puzzling 
behaviors and conflicts that I was seeking. 



31 

Relational Conflict Theory as a Framework 

Relational psychotherapies rest on the premise that repetitive relationship patterns 
derive from the human tendency to preserve the continuity, connections, and familiarity 
of a personal interactional world."'" They recognize that the task of understanding the 
person and helping her disembed from counterproductive interactional patterns may be 
more complex and indirect than cognitive therapy concedes. Like cognitive therapists 
and unlike classical Freudian psychoanalysts, relational theorists regard the person as 
able to consciously choose to change her patterns of thinking and behavior (Mitchell & 
Black, 1995).'" 
Relational Theory, Development, and the Past 

These msights from relational conflict theory promised to explain much of what 
had puzzled me in the learning assistance center. Relational theory acknowledges that 
human beings may proceed as if straightforwardly pursuing conscious goals but asserts 
that, at the subconscious level, they seek to maintain an established sense of self and 
patterns of relationship. In the learning center, I often found students who consciously 
avowed a determination to succeed while they simuhaneously behaved in ways that 
jeopardized that success. The self is not a static entity, however; it simultaneously affects 
and is affected by internal and external realities. As Mitchell notes, the dialectic between 
self-defmition and maintaining connection with others is complex and intricate. He 
theorizes that humans "develop in relational matrices and psychopathology is a product 
of disturbances in both past and present relationships and their interactions" (Mitchell, 
1988, p. 35). Similarly, students' mathematics difficulties may be the product of their 
mathematics learning experiences and relationships interacting with current situations. 



32 

Relational theory does not consider people developmentally arrested by early failures (as 

object relations theorists believe), but rather that they have constricted relational patterns 

that have developed in distorted ways in response to initial and subsequent environmental 

and personal failures. This seems an apt depiction of both the beginning and the 

outcomes of many students' mathematics learning histories. 

These earliest experiences affect subsequent development. Understanding the past 
is crucial... [because] the past provides clues to deciphering how and why the 
present is being approached and shaped the way it is. ... [T]he residues of the past 
do not close out the present; they provide blueprints for negotiating the present. 
(Mitchell, 1988, p. 149, 150) 

My puzzling students' normal mathematical development may have been 
constricted by these negative experiences, and, as a result, subsequent relationships with 
teachers, mathematics, and self became distorted. Their mathematics development had 
also been affected by the effects of their own good and bad choices. The ways they relate 
now to mathematics, to the instructor, and to me, the tutor, provide clues to their past and 
to how to alter their present course. 

Relational theory was offering me a way to understand the development of a 
student's mathematics identity or what I came to call her mathematics self. This theory 
offered me a way to understand how certain experiences and people might have been 
internalized and might affect students' current perceptions of teachers. It also offered me 
a way to understand how loss or change in mathematics and teacher relationships might 
have affected their current relationships to the subject and to teacher. 
Relational Theory and the Student-Tutor/Counselor Relationship 

Mitchell's theory also challenged me with the prospect that a tutor would have to 
take a stance toward the student quite different from the traditional stance. The tutor must 



33 

be prepared to see herself as an integral part of a current relationship with the student and 
be willing and able to use her own feelings and reactions along with the student's 
reactions to her as clues to understanding the student's past. These clues could be used to 
work out with her what to do differently now so as not to reproduce counterproductive 
relational patterns likely to hinder student success. 

Following Freudian psychoanalysis, relational therapists observe and analyze this 
relationship between the counselor and the client to collect key data germane to the 
client's relational patterns. In this framework, a mathematics counselor would also 
observe and analyze this relationship between herself and the student to provide key data 
on the student's relational patterns. Relational therapy is not the same as mathematics 
relational counseling, however. In relational therapy, the interpretation of a client's 
transference of her past relationships into the relationship with the counselor and the 
counselor's countertransference in reaction in her relationship with the client are central 
to the psychoanalytic process. By contrast, although the mathematics counselor's 
conscious awareness and examination of this transference-countertransference dynamic 
will be key to her relational understanding, there is not likely to be time for lengthy 
discussion of this dynamic, nor would the student's need for immediate mathematics help 
or the educational setting make lengthy discussion appropriate. The admittance of 
transference-countertransference as key to diagnosis in mathematics counseling will, 
however, radically change the orientation to the student and her need for mathematics 
support. Relational mathematics support is not only about the student but it is also about 
how the mathematics tutor or counselor experiences the relationship with the student. The 



34 

ways the tutor feels free or constrained in the tutoring relationship become important 
elements in understanding the student. 

Limitations of Relational Conflict Psychoanalytic Theory 
for this Setting 

In embracing relational conflict psychoanalytic theory as the basis of a new 
approach to improving students' mathematics mental health, I had to consider appropriate 
boundaries. It is important to caution myself and the field that adapting relational 
psychotherapies to an educational settmg without proper training is problematic. Even given 
what I now saw to be the appropriate relational emphasis, the sphere of relational history 
exploration needed to be kept limited to mathematics learning settings. Should the tutor 
become aware of connections with more generalized mental health problems during that 
exploration, referral to an appropriate mental health professional would be indicated. 
Exploration of the present tutor-student relationship would also have to be bounded by the 
educational setting. 

Further, any history exploration would need to be conducted while they were 
working on the mathematics. The traditional psychoanalytic leisure to explore at length the 
person's relational past as well as the present therapist-client relationship would not be 
possible or appropriate. Nevertheless bounded strategic engagement of the student in the 
task of exploring and connecting present mathematical behaviors and relationships with past 
experiences for the purpose of freeing her to change these behaviors and relationships, does 
seem appropriate and is what this relational approach requires. 



35 

ADAPTING RELATIONAL CONFLICT THEORY TO HELP 
STUDENTS DO MATHEMATICS 

In order to explore the commonalities I saw between my own puzzling math 
students and the adults for whom treatment with Mitchell's relational conflict therapy 
was applicable, I needed to understand what a bounded and strategic exploration of a 
student's mathematics learning history should entail from a relational perspective. In 
particular, I had to investigate what the findings of the three major relational theories that 
Mitchell integrated into his theory about relationality (self psychology, object relations 
(internalized presences), and attachment theory) could tell me about how a student's 
mathematics relationality might have developed and be expressed in the present. I also 
needed to know about impediments to healthy development along the way, about what a 
student's presenting symptoms tell about that development and current unconscious 
relational conflicts that may impede her mathematical progress. I also needed to know 
ways to improve her mathematics mental health. 

For his conflict relational theory, Mitchell (1988, 2000) drew on (among others) 
key theorists, Kohut (1977) for the self dimension, Fairbaim (1952) for the object 
relations (internalized presences) dimension, and Bowlby (1973) for the interpersonal 
attachment dimension, to explain how each of these relational dimensions differ from and 
complement each other in understanding and helping chents. So these are the principle 
theorists I chose as the basis for my approach. 

In the foUowmg sections, I show how each of the three dimensions of a student's 
relationality around mathematics learning, explained by the Kohut's theory of self, 
Fairbairn's theory of internalized presences, and Bowlby's theory of interpersonal 
attachments, yields a distinctive picture of one aspect of her mathematics identity and 



36 

how she likely developed in relation to the mathematical parenting she received. I show 
how these distinctive pictures complement each other. When taken together, they yield a 
useful picture of her relationality and the mathematics relational conflicts that now 
challenge her, as I illustrate by applying the theories to Janet (see chapter 1 ) following the 
discussion of each dimension. 

The First Dimension: The Self and Mathematics Mental Health 

Self psychology (Kohut, 1977; Mitchell, 1988) looks at adults' relational 
difficulties to discover how their self development might have proceeded and what their 
current self needs are. This perspective provides me a way of understanding the 
mathematics self of an adult student, that is, the core of her mathematics identity. The 
other dimensions then elaborate on interactions with that self The mathematics self may 
be seen as part of a person's academic self, in turn situated in the person's nuclear self. 

According to Kohut (1977), to develop a healthy self the child must experience 
mirroring: unqualified recognition, delight, and admiration from a parent or primary 
caretaker"™'. She also needs the opportunity and indeed the invitation to idealize and 
incorporate into her self a parent image,""" first as part of herself (selfobject) and eventually 
as ideals and values for the self (cf , the superego; Kohut, 1977, p. 185; St. Clair, 1990, 
p. 157). 

If we consider early elementary teaching to be analogous to early parent mg, the 
development of a heahhy mathematical self requires the teacher to initially mirror the 
child's developing mathematical identity, to recognize it, and to dehght in it, much as 
Piaget"™'" (1973) and many cognitive constructivist theorists urge (Windschitl, 2002). 
Simultaneously the teacher provides herself as the mathematical teacher image for the 



37 

student to idealize and to incorporate as part of herself. If early classroom conditions 
facilitate this learning process the student's mathematics self development will likely 
proceed in a healthy manner. 

The elementary teacher's roles in nurturing and facilitating the growth of the 
student's self, in particular her academic self, corresponds in a very real sense to the roles 
of each parent; the mother provides the mirroring and the father provides the parent 
image to be incorporated (see endnotes xvi and xvii). She must reflect back (mirror) to 
the student her mathematics ability, she must allow the student to idealize and internalize 
her mathematics values, and she must provide developmentally appropriate experiences 
(both triumphs and disappointments). The teacher mediates between the formal subject 
matter required by the mathematics curriculum and the informal mathematics the child 
has already developed."'" As the child learns, interactions and connections are made 
among her normal cognitive development, iimate curiosity and exploration, and the 
environment (Ginsburg & Opper, 1979; Piaget, 1967; Vygotsky, 1986). 

For growth to proceed, she must then experience tolerable reality. The self s 
development needs the teacher to occasionally delay or fail to respond immediately to the 
student's demands, thus forcing the self to develop abilities to meet her own demands. 
The student needs to realize that she is not, after all, all-powerful or all-knowing"" (even 
in her teacher's or parent's eyes) nor is her idealized teacher or parent perfectly able to 
meet all her needs. The idealized teacher can no longer be the epitome of rectitude, 
wisdom, or love she initially experienced. She becomes frustrated with the teacher's 
imperfect mirroring and experiences tolerable disappointments with the idealized teacher, 
along with broadening experience that supports her own ability to learn and grow. These 



38 

conditions contribute to the development of a self that integrates a realistic assessment of 
the limits to her own prowess and value with a realistic assessment of the capacity and 
limitations of the idealized teacher or parent. 

The internalized teacher's mathematical values and ideals are integrated as the 
student's own. These internalized values and ideals then provide structure and boundaries 
as the child's own competence develops. When this process proceeds appropriately the 
internal self-structure is consolidated and provides what Kohut (1977) calls "a storehouse 
of self confidence and basic self-esteem that sustains a person throughout life" (p. 188, 
footnote 8). This is the hallmark of a person who exhibits what Kohut refers to as healthy 
narcissism. However, the need for mirroring and permission to idealize continues into 
adulthood. This is a key understanding for a college mathematics counselor to consider. 

If the teacher or parent responds to every demand or fails to respond at all, it 
hinders healthy growth of the nuclear self because the student's own competence does 
not develop in a healthy manner. If a teacher's failure to respond appropriately takes the 
form of overindulgence (e.g., providing too easy tasks and unwarranted praise, having 
high expectations with little pressure for the student to meet them) the student's 
grandiosity is not appropriately challenged by reality and she develops what Winnicott 
might call a false mathematics self (cf St. Clair, 1990). Her self-esteem remains low 
because her competence does not develop appropriately but a defense is likely in the form 
of unrealistic over confidence. She "knows" she can achieve if she wants/tries to. On the 
other hand if the teacher's response is in the form of chronic neglect'™' (e.g., expecting 
little when a student falters or seems slow to grasp concepts and subsequently ignoring 
her need for challenge, tracking into low level tracks) she fails to see herself mirrored in 



39 

the teacher and her mathematics self fails to develop. In the extreme this may result in 
what almost feels like the absence of a mathematics self (cf Cara in Knowles, 2001). Her 
competence and therefore her self-esteem remam low as is true for the overindulged 
student, but the neglected student's defense is likely to be different, in the form of 
unrealistic under confidence. She is sure that she cannot succeed. 

This study of self development allows me to see that a student whose mathematics 
self is vulnerable because it is underdeveloped or undermined has likely developed 
defenses (typically under or overconfidence and accompanying avoidance behaviors) to 
protect this self from fiirther damage. Although her conscious goal is success in her 
mathematics course, she likely acts in ways that jeopardize that goal. Her self-esteem is 
compromised or low and she may have little underlying belief that she can succeed. Her 
unconscious goals are in conflict with her conscious ones and she remains embedded in 
her familiar patterns of relationship with self (cf Mitchell, 1988). 

The Second Dimension: Internalized Presences — Objects Relations and 
Mathematics Mental Health 

Object relations theory principally focuses on the person's interior relational 

world. This world is conceptualized as the person's self in relationship with internalized 

and altered others (objects of the persons' feeling and drives), with split-off parts of self, 

and with external others (objects). Whereas the focus of self psychology is on the 

development of structures of the self, the focus of object relations is more on how early 

interpersonal relationships are internalized and on how the irmer images of the self and 

the other (object) are formed and shape perceptions and ongoing relationships with real 

and internalized others (Fairbaim, 1952; St. Clair, 1990). From this perspective, a 

student's internal reality is peopled by objects and selfobjects that affect her mathematics 



40 

self and the way she perceives external reality, in this context, the current mathematics 
instructor and course. 

If parenting is experienced as threatening enough to the self, bad internalized 
presences are formed, creating internal conflict that distorts the person's perceptions of 
present reality. Fairbaim (1952) contends that "internalized bad objects are present in the 
minds of us all at deeper levels" (p. 65) and the degree to which they negatively affect us 
in the present depends in part on how bad we experienced the original external other 
(object) to he.'^ 

In the elementary classroom, a student cannot get away from the teacher and, in 
fact, needs her. If the teacher humiliates the student or those around her, abuses her 
verbally and or even physically, """ or otherwise creates a classroom environment that the 
child experiences as unsafe, the child may cope with what feels like an intolerably unsafe 
situation by holding the teacher to be good (right) and internalizing the bad part of the 
teacher in order to feel safe, at least externally. She may then handle her now intolerably 
unsafe internal situation by the defense of repressing the bad internalized object (the 
teacher) or by a defense that Fairbaim (1952) calls the "defense of guilt" or "the moral 
defense" (p.66). That defense is accompUshed thusly: The student or child is in a 
situation where she feels surrounded by bad objects. Because this is intolerably 
frightening, she converts this into a new situation where her objects (parents, caregivers, 
teachers) are good and she herself is bad. A student or child who has suffered abuse or 
neglect typically refiises to characterize the parent as bad, but is quick to admit that she 
herself is bad. 



41 

It is not only students who have been abused who see themselves as bad and feel 
shame and guilt; neglected students also feel shame for their deficiencies. The shame of 
both abused and neglected students seems related to a sense of nakedness or sin, as if 
their internalized mathematics object world is dominated by mathematics in the form of a 
judgmental superego*""^ or by a bad mathematics teacher, threatening to unveil the 
deficiencies of the vulnerable trying-to-hide mathematics seir"^ and the result is a 
fearful, beleaguered mathematics self (cf Buxton, 1991). 

A bad teacher presence (or object) assaults or conflicts with the student's 
developing mathematics self and sabotages future relationships with teachers, even good 
ones. What is pertinent for understanding the adult is not so much what actually occurred 
between the teacher and child'""' but how the child experienced the mathematics teacher 
and mathematics, how she internalized them, and how she as an adult now experiences 
them. The student's initial transference relationship with the mathematics counselor and 
the instructor is likely to reveal much about such presences. If her internalized good*"^" or 
bad presences (especially internalized bad mathematics teacher-objects) are not brought 
to consciousness and released, they may continue to control the present-day learning 
relationships in a negative way. 

Whether the student's efforts to deal with internalized bad teacher presences have 
involved repression of bad teacher presences, moral conversion into herself being bad, or 
another defense, when she enters the current classroom these unconscious forces are 
activated and internal conflict develops between resignation to her mathematical badness 
and her motivations to succeed in the class. Internalized presences may be so prominent 
that they take precedence over current reality; the student may relate to the present 



42 

teacher as if she were in the original classroom. Conflicts arise when this mismatch 

between her internal and external reality negatively affects her progress in the course. If 

these conflicts are not resolved satisfactorily her desire to succeed or even survive in the 

course may be thwarted. 

The Third Dimension: Interpersonal Relational Attachments and 
Mathematics Mental Health 

The exploration of object relations gave me insights into how a student's internal 
relational world might be configured and might now be affecting her. Attachment theory 
promised to give me insight into the development, significance, and challenges of her 
external interpersonal relationship dimension of relationality. In particular, attachment 
theory examines the ways the person forms ongoing relationships with significant 
persons in her life and work (Bowlby, 1965, 1982). Often her tendency towards 
dependent, detached, ambivalent, or self-reliant relationships will provide clues to the 
security of her early relationships and her subsequent experiences of loss or change in 
those relationships. The extent to which a college student seeks the help she needs when 
she needs it from her instructor, learning assistance personnel, or other suitably 
knowledgeable person has been found to be an important factor in her success (cf 
Downing, 2002, Zimmerman & Martinez-Pons, 1990). The student's established 
attachment relational patterns may determine whether she is likely to make contact at all 
with those who could help her, and if she does, how she proceeds to do so. 

Attachment theorist John Bowlby (1973, 1982) and his colleagues found strong 
evidence of a child's instinctive need for secure attachment to a particular parent figure. 
The attachment-caregiving bond developed between child and mother figure'""'" is seen 
as crucial to child's survival and forms the basis for any future attachment relationships 



43 

the child develops. The type of attachment achieved by the child varies according to the 
type of caregiving the mother figure provides the child. Most important factors in 
mother's caregiving are her responsiveness to the child's signals (e.g., crymg) and the 
extent to which she initiates social interactions with her baby (Bowlby, 1982, pp.3 12-3 18, 
referring to studies by Schafifer & Emerson, 1 964, and a study by Ainsworth, Blehar, 
Walters, & Wall, 1978). Secure attachment is achieved when the caregiving by the 
mother figure is characterized by being sufficiently available and responsive. The mother 
figure becomes the secure base from which the child can move out and explore her 
world, but return to for comfort and reassurance in times of distress. 

Researchers have found that a child's insecure attachments can be explained by 
the caregiver's behaviors towards the child. The caregiving that detached insecure 
children receive is consistently detached, with the mother figure rarely responding to the 
child's expressed needs and rarely herself initiating positive interaction with the child. 
Children whose insecure attachments are ambivalent, alternating between demanding 
contact with their mother figiire and resisting, receive inconsistent or conflicted 
caregiving that the child finds unpredictable in its quantity or quality or both (Ainsworth, 
Blehar, Walters, & Wall, 1978). Another insecure pattern, disorganized anachmsrA, is 
characterized by fear of the caregiver or of her leaving or loss (Jacobsen & Hofinarm, 
1997). '"" 

The peculiar mark of a securely attached child is her exploratory, adventurous 
behavior, as long as she is assured of the availability of her attachment figure if needed. 
By contrast, the insecurely attached child is preoccupied by frequently thwarted attempts 
to avoid further separations from her attachment figure; she stays close and is afraid to 



44 

explore lest she be abandoned or punished, or she tries to meet her own needs, distancing 
herself from her detached attachment figure. The secure person's behaviors lead to 
learning; those of the insecure person's tend to inhibit it. Students' academic competence 
through adolescence is also likely to be positively related to the security of their 
attachments (Jacobsen & Hofmann, 1997). These outcomes are not unexpected. Many 
educational researchers have demonstrated that the student's learning is dependent on her 
investigating and interacting with her environment (cf Dewey, 1903; Piaget, 1973; and 
others). 

The subsequent ability of a person who has developed insecure attachments to 
form relationships with others will be negatively affected and may be permanently 
marred. By analogy, early experiences in a mathematics classroom where the teacher 
does not understand or respond to the child's need for cognitive and emotional support, 
challenge, and latitude for exploration may lead to a sense of insecurity and difficulty 
with trusting the next teacher and subsequent mathematics material. Her beliefs and 
behaviors may resemble anxious learned helplessness on the one hand or mistrusting 
independence on the other. 

People whose primary secure attachment relationships have not been unduly 
disrupted usually develop into adults who form secure attachments. They are what 
Bowlby (1973) calls truly self-reliant, "able to rely trustingly on others when occasion 
demands and to know on whom it is appropriate to rely" (p. 359).'°°' Because these 
people are confident that an attachment figure will be available to them when they need it 
(a secure base), they are much less predisposed to intense or chronic fear than a person 
who does not have that confidence (Bowlby, 1973; Sable, 1992; Weiss, 1991). They are 



45 

more resilient and able to negotiate difficult circumstances more successfully than those 
whose early attachment bonds were insecure (see also Werner and Smith, 1982). In 
contrast, adults whose attachment bonds were insecure or whose secure attachment bonds 
were traumatized are likely to establish insecure attachments and have difficulty in 
withstanding life's or the mathematics classroom's difficulties in a healthy manner. 
Attachment bonds they form as adults are likely to be anxious, ambivalent, detached, 
disorganized, or a combination of these.'"™ 

The teacher-student relationship, especially in the early years (generally through 
third grade), is a type of attachment/caregiving relationship more than a relationship of 
community'™'" although the teacher is not a substitute parent for her students. Even in the 
early grades, there are important distinctions between parent figure roles and the teacher 
roles. In particular, the teacher's relationship with the child should be characterized by 
appropriate responsiveness and caregiving without the intense emotional involvement of 
parental attachment (Katz, 2000).'™"" As the student gets older, the focus of the teacher's 
"detached concern" care becomes a narrower one with more emphasis on providing an 
academic secure base and less on emotional involvement (cf endnote xxxiii). A tutor or 
learning counselor role is perhaps an intermediate one, with more emotional involvement 
and partiality than is generally appropriate for a teacher. In a small college, for older 
adult students, as well as for adolescent/young adult students, the power differential in 
the 20- to 30-student classroom between the instructor and students and its similarities to 
classrooms of the past can activate established teacher relational patterns that are more 
akin to adult attachment than community relationships. 



46 

An unsafe or unsupportive classroom environment can certainly cause or 
contribute to the development of insecure attachments to teacher or mathematics or both 
(Dodd, 1992; Fiore, 1999; Jackson & Leffmgwell, 1999; Knowles, 1996; Mau, 1995; 
Tobias, 1993). Students' subsequent avoidance of mathematics has been linked with 
ambiguous and unsupportive classroom envirormients (Patrick, Turner, Meyer, & 
Midgley, 2003). 

It is not only student-teacher attachments that are affected by the way the teacher 
manages the learning envu-onment. Student-mathematics attachments are also affected. 
U.S. elementary teachers are likely to lack a secure base™"^ in the arithmetic they teach 
(L. Ma, 1999), and those with insecure attachments are less able to provide secure 
attachments to those in their care (Ainsworth, 1989; Bowlby, 1980). By extension we 
may assume that in these mathematics classrooms, students' attachments to the 
mathematics itself are vulnerable. Classrooms where the instructor provides either too 
much or too little conceptual mathematics structure may inhibit students from making 
healthy attachments to the mathematics. Likewise, teacher-as-authority mathematics 
classrooms may also hinder healthy student attachments to the mathematics. Instead 
students may develop an anxious attachment to mathematics that undermines their 
confidence in feedback they get from working with the mathematics, and may keep them 
unhealthily dependent on the teacher for decisions about whether they are proceeding 
correctly. Confirming this, Skemp (1987) considers unhealthy dependence on the teacher 
to be one of the chief drawbacks of an overly procedural approach to teaching 
mathematics. 



47 

When the teacher's attachment to the mathematics is insecure, she is likely to 
cling anxiously to procedures, not daring to explore or question, fearful that her 
procedural grasp of the mathematics may be lost. She is less able to entertain students' 
queries (much less, encourage their exploration) and is likely to respond with censure to 
correct or logical approaches that differ from her grasp of the mathematics (cf Corwin, 
1989; L. Ma, 1999). But if the procedural teacher has a secure mathematics base, the 
prognosis for students' secure attachment to the mathematics is better even if it is 
hampered by lack of encouragement to explore the mathematics for herself and construct 
her own understandings with the teacher as guide. It is not only procedural transmission 
pedagogical approaches that may jeopardize students' attachments to mathematics. 
Students in laissez-fair classrooms are likely to lack a mathematical secure base™" and 
even those in constructivist problem-solving classrooms may feel anxious and abandoned 
xmless they are oriented to expect uncertainty as part of the problem-solving process and 
appreciate the real availability of a mathematical secure base.™"' 

Students' well-developed secure attachments to teachers and to mathematics can 
be disrupted by a negative experience with a teacher or encountering a type of 
mathematics or teaching style that result in a poor grade or failure. How well a person of 
any age negotiates loss and avoids distortion of psychological development depends on 
three factors.™"" The third factor: the continuity and quality of her relationship with 
other primary attachment figure/s after loss or separation (Bowlby, 1980) is of particular 
importance to a mathematics counselor working with students. It seems to me that a 
counselor would fmd it easier to help students who had at some time experienced secure 



48 

attachment to mathematics to reattach to it than those who had never felt securely 
attached to mathematics. 

Change can also disrupt mathematics and teacher attachment relationships and 
without support to negotiate the change students may remain stuck in a natural resistance 
that could jeopardize their future success. Even when changes can be seen by outsiders to 
be for the good, people are likely to resist or even reject™"'" those that cause disruptions 
to their attachments to relationships and circumstances. This may help to explain why a 
student repeating a course taught by a different instructor may resist approaches that are 
different (and often preferable), even though the student initially failed with the 
approaches she clings to. When students find themselves in a classroom whose approach 
is different from the ones they are used to they are likely to experience what Marris calls 
a "conservative impulse" to resist changes that call into question their familiar ways of 
doing mathematics (cf Bookman & Friedman, 1998). I realized that helping these 
students recognize and work through their resistance might free them to benefit from the 
new course situation, but that would only be possible if I or the instructor or both 
provided a secure base and the students could attach to it in the new situation. In order to 
successfiiUy resolve the effects of loss or change experienced as loss, a person must work 
through a grief process™"" to "retrieve the meaning of the experience and restore a sense 
of the lost attachment that still gives meaning to the present" (Marris, 1974, p. 147, 149). 

If a student has developed attachment patterns to mathematics teachers or tutors 
that are characterized by investing either too little or too much reUance in the teacher or 
tutor, their success or at least growth in mathematics learning may be compromised. They 
face a likely conflict between maintaining their familiar but coimterproductive attachment 



49 

patterns and their willingness to risk trusting a relative stranger enough (e.g., the learning 
counselor who is a mathematics "teacher") in order to attain a healthier balance between 
their responsibilities and getting the appropriate help they need. 

Application of Relational Conflict Theories to the Case of Janet 

When I look again at Janet (see chapter 1) through the lens of relational conflict theory, it 
seems likely that her first grade teacher's failure to mirror her already existing 
mathematics ability and the teacher's developmentally inappropriate prohibition of the 
use of concrete models to build understanding and provide transitions to internalized 
knowledge had impeded the development of her mathematics self-esteem which is the 
basis for a sound sense of mathematics self. This teacher had pushed underground her use 
of fingers as a transitional object"' so that she had never developed beyond needing them 
(at least emotionally) and still used them for security in an insecure world, despite the 
risk of embarrassment, or worse, shame. This seems to have resulted in her seeing herself 
as bad (at mathematics) because the teacher had to be good (or at least correct) in her 
judgment of Janet and in her actions. It had thus distorted her sense of her mathematics 
self. 

Despite this inauspicious start it was now apparent that Janet's mathematics 
competence had developed though it remained undermined and her self-esteem remained 
low. She expressed her low self-esteem in an underconfident, resigned (perhaps 
depressed) determination to proceed, with little hope of feeling secure in her grasp of the 
material. It seemed that Janet had failed to develop initial secure attachments to 
mathematics or to mathematics teachers and now her relationships with those from whom 



50 

she sought help seemed wary; she hid from them her shameful and illegitimate 
techniques, expecting ridicule. Traditional understandings of Janet's affective problems 
couched in terms of mathematics or testing anxiety and counterproductive beliefs related 
to helplessness and her other unhelpfiil approaches, may now be seen as clues to her 
underlying relational issues. Thus these affective problems could now be seen as 
symptoms rather than causes of her difficulties. What was sound and healthy about her 
affective orientation to mathematics learning could likewise be seen as symptomatic of 
aspects of sound mathematics self-esteem. 

Janet's mathematics cognitive knowledge, conceptions, and approaches can be 
seen in context of and as outcomes of her mathematics relational history. Her current 
patterns of mathematics learning and production can be understood as symptomatic of her 
underlying sense of mathematics self. 

Mathematics Parenting of Janet from the Three Perspectives 

Each of the three relational perspectives gave me insight into parenting and 
analogously into teaching as parenting. On reflection, two considerations stood out. 

First, although it might appear that each says basically the same things about the 
essentials and processes of early teacher-parenting, in fact that is not the case. Each 
perspective does give different insights. Taken as a group of theories they are, as Mitchell 
(1988, 2000) has shown, complementary with intersecting areas of interest. 
Understanding the different related conflicts an adult might be experiencing, depending 
on the dimension, promised to yield much in effectively diagnosing a student's 
mathematical challenges. Second, I realized that Winnicott's (1965) concept of good- 



51 

enough mothering or parenting is a unifying concept that apphes in each perspective and 
could be especially useful in my work with college students. 

Different perspectives on teacher-parenting. Looking again at Janet, self 
psychology's perspective would prompt me to examine her confidence level in relation to 
her mathematics achievement to gauge the state of her mathematics self-esteem. When I 
found that she expressed unrealistic underconfidence given her achievements I would 
speculate that her early (and subsequent) teacher-parents failed to adequately nurture her 
developing mathematics self As her counselor I would explore this speculation with her 
and look for ways to re-parent her mathematics self now. I would find and help her 
recognize and receive as her own her existing competencies and understandings (through 
mirroring). I would expect and push the development of fiirther competencies and 
understandings by initially allowing her to idealize and rely on me but progressively 
challenging, frustrating, and disappointing her so that she would become more and more 
reliant on her own competent self 

An object relations perspective would lead me to clues to Janet's internal 
mathematics relational life. I would now look for evidence of internalized teacher 
presences, her use of repression as a defense, her moral conversion of herself as bad to 
keep her bad teacher good, or other unconscious defenses in the face of her experiences 
of traxuna in relation to teacher-parents (or parents as teacher/tutors). The teacher- 
parenting central to this perspective is what the child experienced as traumatic. The 
discrepancy between how she now relates to her current teacher and/or tutor, and how 
they are in reality, is a clue to the influence of internalized realities. Applicable 
counseling interventions would involve the counselor's providing herself as an especially 



52 

"good" teacher-parent and helping the student to become consciously aware of the 
"goodness" of the current classroom teacher so that she can safely let go of detrimental 
internalized teacher presences and incorporate instead the "good" teacher and the "good" 
counselor. 

From the perspective of attachment theory, I would notice Janet's occasional 
wariness, and her intermittent dependence on me. She had little apparent relationship 
with her classroom teacher, and lacked confidence in how she did mathematics even in 
the face of good results. Taken together, these seem likely indicators of insecure 
attachments to mathematics teachers and to mathematics. These attachment patterns 
point to failure of past teacher-parents to provide a teacher secure base and a secure base 
in mathematics. 

Early teachers may have provided a secure enough base only to have that 
disrupted by later teachers. In the case of Janet it seems that she had certainly not had an 
early mathematics teacher who offered her the cognitive and emotional support, 
challenge, and latitude for exploration that she needed to develop secure attachments to 
teachers or mathematics. As a consequence, she had developed ambivalent patterns of 
relationship. As a counselor I would provide myself as a consistent, safe secure base, 
nevertheless challenging Janet, and pushing her to move away and explore and make 
mistakes so she could experience returning to the base to find it secure and accepting. 

While all three dimensions of a student's relationality should be the objects of a 
counselor's curiosity, it is likely that any particular student's mathematics mental health 
problems might be based more firmly in one of the areas than in the others at the time of 
the brief counseling. Mitchell (2000) shows that as longer-term relational conflict therapy 



53 

proceeds and difficulties in one dimension are resolved, difficulties that emanate from 
other dimensions will likely emerge to be dealt with. 

Good-enough teacher-parenting. A good-enough mother, like a good-enough 
teacher, provides sufficiently for the child to get a good start in Hfe by adapting 
adequately to the child (or student) and her needs (St. Clair, 1990). This is an 
empowering acknowledgement of the inevitable imperfections in parenting or teaching 
that are nevertheless tolerable (or even necessary, within appropriate limits), for the 
healthy development of the student's self Even if a student had experienced mathematics 
classrooms as bad, had low mathematics self-esteem, and viewed her prospects as bleak 
in the current class, I believed it was likely that we could find instances of good-enough 
teaching and understanding so that some of the bad could be appropriately reinterpreted 
and re-experienced as good-enough, providing bases for hope and progress. I use good- 
enough to refer to the present, not only to current teaching and tutoring/counseling 
conditions for the student but also to her process, progress, and outcomes. If the student 
and I can let go of a perfect-or-nothing requirement and instead embrace good-enough for 
ourselves, each other, and the teacher, we could perhaps make good-enough progress and 
the student could achieve good-enough success. 
Janet 's Relationality Summary 

If I had offered Janet mathematics relational counseling, it would have involved 
the kind of mathematics tutoring designed to help her recognize, draw on, and develop 
her mathematical understandings and strengths while simultaneously attending to, 
processing and dealing with her affective and cognitive symptoms of difficulty. A newly 
developing self-esteem would likely have led to changes in her ways of seeing her 
mathematics self, improvement in the way her internal mathematics world was 



54 

configured and in repaired attachments to the tutor, the teacher and mathematics. This 
new freedom from formerly constricting relational patterns could lead to progress in 
alleviating her negative symptoms, maximizing her mathematics potential, and achieving 
good-enough success. 

CONCLUSION: RELATIONAL CONFLICT THEORY AS A BASIS FOR 
MATHEMATICS COUNSELING 

Relational conflict theory had given me a way to explore how a student's self- 
esteem and her beliefs, habits, ways of relating, and behaviors may be related to each 
other. How the three dimensions of her relationality interact (her self, her internalized 
presences, and her interpersonal attachment patterns), and the relational patterns she 
employs to express that interaction give me the understanding I sought. Relational 
theories point to some ways to identify and resolve her central conflict (Luborsky, 1 976; 
Luborsky & Luborsky, 1995) and free her from the counterproductive relational patterns 
limiting her progress. My adaptation of Mitchell's (1988, 2000) relational conflict theory 
had given me a new way of looking at the student and at our relationship. I determined 
that it was an approach that could include the insights and best practices of traditional 
mathematics tutoring within a broader and deeper relational coimseling framework (see 
Appendix A for a chart summary of the proposed mathematics relational counseling 
approach). 

In the next chapter I show how a relational conflict counseling approach could be 
used appropriately and integrated with best practice mathematics tutoring in the setting of 
the learning assistance center. 



55 



' In even numbered chapters, 1 use "she," "her," and "hers" for the third person generic singular. 

" Davidson (1983) found strong linlcs to hemispheric preference and clearly defined analytic (her 
Mathematics Learning Style I — left brain) and global (her Mathematics Learning Style II — right brain) 
learning styles in terms of students' mathematical behaviors and approaches. Although these learning styles 
have not been found to be directly related to mathematical achievement, Krutetskii (1976) found that 
students who had a strong learning style preference and a relative inability with their other mode, found it 
difficult, if not impossible, to begin a problem using the other mode's approach. Thus being forced to 
approach problems using another's preferred approach greatly disadvantages these students. Learning 
flexibility, however, can strengthen performance. 

Students with an analytic/Mathematics Learning Style I use predominantly verbal-logical methods to 
solve problems, use deductive approaches, and prefer to follow step by step procedures. Krutetski labels 
them analytic. Students vnth a global/Mathematics Learning Style II use predominantly visual-pictorial, 
inductive reasoning methods to solve problems and may know the answer to problems without being able 
to explain how they arrived at it. Krutetskii labels them geometric. I hesitate to use Krutetskii's term 
geometric because although he and others have generally found some relationship between success in 
geometry and this visual processing right-brain preference learning style, analytic learners also achieve 
success in geometry courses. Success in geometry is therefore not a clear indicator of a global learning 
style. 

'" To explain the difference between procedural and conceptual teaching, 1 offer the following example. 
Teaching the fectoring of a trinomial X" + 7X + 12 by finding fectors of 12 that sum to 7 and putting those 
into (X + )(X + ) gives a procedure that may be memorized but probably neither linked with prior learning 
nor generalized to a more complex fectoring problem such as 3X^ + 16X - 12. Thus a new procedure must 
be learned for this one, such as breaking apart 16X in a way that the coefficients multiply to equal -36 (i.e., 
3 x-12); then fector by grouping. Conceptual mathematics, as the term infers, is taught and learned as 
concept-based processes that put less load on rote memorization and are more easily generalizable to new 
more complex though related problems. For example the fectoring of the trinomial X' + 7X + 12 using a 
conceptual approach might be linked with the earlier process of multiplying binomials X + 3 and X + 4 
(and the geometric relationship of multiplying the length and width to get the area of a rectangle), and still 
earlier distributive explorations of operations on number using two digit by two digit multiplication (also 
area of a rectangle). The relationship between multiplication and division would be explored and 
equivalence of division with fectoring made clear. Finding then that (X +3)(X + 4) = X"+ 7X + 12, and 
relating this with the idea that 23 x 24 = 400 + 60 +80 + 12, that is, 20^ + 3x20 + 4x20 + 3x4, leads 
students to explore the relationship between the 3 and 4 in the fectors and the 7 and 12 in the product and to 
fiirther explorations and discoveries that are applicable to other problems. 

" That is, not specifically related to a mathematics course the student is currently taking. 

" Equivalent to what Hiebert (1986) and others refer to as "procedural" when they discuss mathematical 
understanding. 

" Equivalent to what Hiebert (1986) and others refer to as balanced "conceptual" understanding with the 
requisite procedural knowledge. 

"' Measured on Richardson and Suiim's (1 972) Mathematics Anxiety Rating Scale (MARS). 

"" Hypothesis testing of a feulty belief might involve having a student viiio believes that people who do 
well in math just see it immediately and do not need to work, interview some high achieving math students 
who do have to work hard to understand and achieve. 

" A cognitive restructuring exercise might involve having a student who sees herself a "bad" at math and 
points out as evidence the errors on her quizzes and tests and any overall poor grades, develop the practice 
of noticing instead not only the questions she did correctly but also her sound thinking even in the 



56 



questions she got wrong. If she combines this new practice with seeing the Hnic between insuificient or 
inefficient preparation and her poor results, if she changes her preparation, and if she begins to see a change 
in her results, her overall approach should change and her perception of herself doing math should also 
improve. She has experienced cognitive restructuring. 

" A person whose locus of control in a mathematics learning setting is external is likely to attribute her 
achievement outcomes to factors that she feels she cannot change or control such as luck, the teacher, the 
tutor, the weather, her health at the time, her lack of intelligence in mathematics (that she believes is a fixed 
trait), etc. On the other hand a person whose locus of control in a mathematics learning setting is internal is 
likely to attribute her achievement outcomes to factors that she feels can change or control such as her own 
effort, her intelligence in mathematics that she believes can improve, getting the support she needs, 
strategic planning for tests, etc. 

" A student whose motivation for achievement is primarily performance is focused on passing or getting a 
particular grade rather than on understanding the material. In contrast a student with learning achievement 
motivation is primarily focused on understanding the material. 

^ Others have looked into the unconscious and psychoanalytic symbols found in mathematics and 
discovered there the roots and explanations for panic, aversion, and defenses against mathematics (of., 
Nimier, 1993; Tahta, 1993). 

"" Object is used hwe in contrast with subject. In other words the object is the "other" in contrast with the 
subject, which is the "self" According to object relations theory, early significant others become 
internalized in various healthy and unhealthy ways as internalized presences that influence how the person 
relates to others subsequently. I prefer to use the terms "other" or "presence" ratha than "object" because 
"object" now has somewhat negative connotations implying a sense of persons as things. 

'^ [PJsychopathology, in its infinite variations, reflects our unconscious commitment to stasis, to 

embeddedness in and deep loyalty to the familiar... we experience our lives as directional and linear, 
but like Penelope. . .we unconsciously counterbalance our efforts, complicate our intended goals; 
seek out and construct the very restraints and obstacles we struggle against. (Mitchell, 1988, p.273) 

"" Freud's view is that a person's choices are largely determined by unconscious instinctual drives and 
forces outside of ha conscious control. Recognizing that the person, in contrast, is responsible for her 
choices and actions, implies that helping her become conscious of her hidden motives should provide both 
more insight into puzzling behaviors and also the possibility of modifying hidden motives in light of 
conscious goals. The consistent relationship between academic/mathematics achievement and locus of 
control (see Nolting, 1990, McLeod, 1992) is pertinent here. Students who fail to see their own 
responsibilities in achieving success in a course, holding others or external fectors responsible instead, 
consistently achieve less well than those who own that responsibility (internal locus of control). 

"^ Usually seen as the mother although the role rather than the gender is the central factor. Winnicott 
(1965) conceives of the function of mother as providing experiences to make possible a sense of 
authenticity and reality; that is, to provide "good-enough" mothering that leads to "maturity and the 
capacity to be alone ... [and] a belief in a benign environment"(p.32). Kohut (1977), in expanding 
Winnicott's findings, also sees that the child's nuclear or core self arises as the result of the interplay 
between her innate potentials and the responsiveness of the adult selves which the child internalizes as parts 
of herself 

'"^ Usually seen as the fether although the role rather than the gender is the central factor. 

"™ Piaget (1973) contends that children's intellect develops primarily through self-directed activity, both 
physical and mental. He asserts that all learning is "of a constructivist nature ... affirms a continuous 
surpassing of successive stages. . . leads to placing all educational stress on the spontaneous aspects of the 
child's activity... The basic principle of active methods will have to draw its inspiration from the history of 



57 



science and may be expressed as follows: to understand is to discover, or reconstruct by rediscovery, and 
may be complied with if in the future individuals are to be formed who are capable of production and 
creativity and not simply repetition" (p. 1 0). 

"^ When a student tells a learning counselor that she has never been "good" at mathematics, even in first or 
second grade, we must question her early experience of teacher-parenting. Research findings assure us that 
barring a severe specific learning disability, developmental delay, emotional disturbance, or physical or 
emotional abuse or deprivation, the average intelligent child is mathematically capable when she enters 
school (Caufield, 2000; Hawkins, 1974; Kamii & DeClark, 1985; Kunzig, 1997). She has all she needs to 
explore and learn developmentally appropriate number and operation concepts and their symbolic 
representations, along with applications in solving problems based in her real world. If she does not 
remember experiencing success it suggests many possibilities but most likely is that her early teachers did 
not mirror her developing ability to do mathematics in a her ovm way or provide appropriate challenge and 
frustration to promote her competence. 

"' That is, her grandiose (to her, all-knowing and all-powerful) self is challenged and modified by reality. 

™ Kohut (1977) maintains that it need not be specific traumatic events, but rather the chronic absence of 
the parent's empathic responses to the child's need to be mirrored and to idealize that may lead to 
pathology of self in the adult (p. 187). 

'"" Fairbaim (1952) further asserts that 

Whether any given individual becomes delinquent, psychoneurotic, psychotic, or simply 'normal' 
would appear to depend, in the main, on the operation of three fectors: (1) the extent to which bad 
objects have been installed in the unconscious and the degree of badness by which they are 
characterized, (2) the extent to which the ego is identified with the internalized bad objects, and 
(3) the nature and strength of the defenses which protect the ego from these objects, (p.65) 

"^ For example, rapping knuckles, pinching, hair pulling. Although these activities are illegal in the U.S., 
they continue to be practiced, particularly in poorer communities where parents may feel less empowered 
to challenge school practices. 

'°^'' Or what Fairbaim (1952) calls the "internal saboteur." 

""" This is a plausible explanation for what I have found to be the puzzling phenomenon of adult students 
apologizing to me when they find or 1 point out an error in their work (especially an error in arithmetic) 
saying, "I'm sorry," almost as if they have committed a sin and deserved punishment. Evidence of their 
badness has been revealed and the effect is invariably shame. Are they ashamed because their early 
teachers shamed them when they made such mistakes? Or are they ashamed that as adults they have 
revealed incompetence at something a young child should be able to do? 

"^ It is important here to distinguish between what actually happened, that is, what the teacher did in the 
classroom, and how the adult student now remembers cognitively, affectively, and overall relationally 
experiencing it. The forma- is impossible to verify and is not as relevant as the latter which is what is 
affecting her now. 

""^ Fairbaim (1952) referred to the internalization of good objects only in terms of the super-ego and the 
development of principles and values much as Kohut (1977) saw the healthy modification of the parent 
image. Bad objects on the other hand were internalized and interacted with the ego (operating part of the 
self) causing conflict and splitting, that is, trouble when they were repressed or otherwise dealt with 
internally. 

'°™" Although this person is most often the child's biological mother, others, including the father or other 
relative or unrelated person may be the mother figure for the child (Bowlby, 1982). 



58 



""^ There is some evidence that children of mothers who themselves suffer from unresolved attachment 
trauma or loss are likely to develop this disorganized attachment (Main & Hesse, 1990). It seems that 
many of the attachments formerly identified as ambivalent may be more accurately identified as 
disorganized. 

^ Bowlby distinguishes the concept of "self-reliant" from that of "independent," pointing out the cultural 
stereotype of an independent person as one who relies only on self and repudiating or not needing the help 
of others (Bowlby, 1973). Bowlby's concept of self-reliance is closely linked with Werner and Smith's 
(1982) concept of resilience and Lillian Rubin's (1996) concept of transcendence. Werner and Smith found 
that a key to a child's resilience under difficult circumstances was her significant relationship with an 
accepting, approving, and challenging adult. Likewise, Rubin found that adults she studied who had 
transcended abusive childhoods had all had such a relationship with an adult as a child, that had enabled 
them to survive emotionally and become self-reliant adults themselves. 

'°™ Disordered adult attachment behavior patterns linked with early insecure or interrupted attachment 
relationships include 

1 . Anxious attachment, characterized by over-dependence or clinging and severe separation anxiety, 
thought to be linked to threats of abandonment by the childhood mother figure or to her forcing the 
child to take on the caregiving role, 

2. Insistent self-reliance, characterized by an apparent lack of any need for relationship or assistance, 
thought to be connected with early rejection or prohibitions on expressing emotions or needs as a child, 

3. Insistent or anxious caregiving, typified by exclusive formation of one-sided relationships in which 
she is always the caregiver, thought to have developed from the experience of the mother figure's 
expecting the child to mother her. 

4. Detachment, characterized by emotional detachment and an inability to form stable bonds, stemming 
from separations from the mother figure that were severe or prolonged (Bowlby, 1980; Sable, 1992). 

'""^ Not all important relationships are attachment relationships. Attachment relationships, even for adults, 
are characterized by "proximity seeking [to the attachment figure], secure base effect, and separation 
protest" (Weiss, 1991, p.66). They contrast with community relationships "that link individuals to networks 
of fellow workers, friends, or kin" (p.68) that also likely characterize peer relationships in a college 
classroom. 

'°™" Katz (2000) describes seven role dimensions where there are important distinctions between parenting 
and teaching. They are: scope of fimction, intensity of affect, attachment of adult to child, rationality, 
spontaneity, partiality, and scope of responsibility (p. 1 1 ). Of particular interest here are the dimensions of 
attachment and partiality. Katz proposes that whweas parenting should be characterized by optimal 
attachment with the child (essentially, secure attachment, appropriate caregiving), teaching should be 
characterized by optimal detachment, or "detached concern," to use Maslach and Pines' (1977) term, 
characterized by appropriate responsiveness and caregiving without the intense emotional involvement of 
parental attachment (whether and how this optimal detachment is to be achieved may be related to the 
teacher's ovm attachment history and to resolution of fransference and countertransference issues) With 
regard to the partiality dimension, the parent's role is to be partial, biased towards her child; the teacher's 
role is to be impartial, unbiased in relation to any one child but biased in her relationship with the class as a 
whole. 

These teacher and mother roles work best for the child if they are age-appropriate and complementary. 
The teacher "is seen as wiser [academically] and sfronger [in relation to classroom management] and 
therefore able to be protective at times when the self seems inadequate" (Weiss, 1991, p.68). 

"""'' What Ma (1999) calls "a profound understanding of fijndamental arithmetic." 

'°°" The effects of a laissez-feire classroom may be detrimental. Poorly planned discovery learning 
situations where student are expected only to explore without knowledgeable and strategic teacher guidance 



59 



and support are unlikely to result in much mathematical learning (cf. G. Hein, personal communication, 
September 1994). They are very likely to result in knowledge base gaps and insecure attachment to 
mathematics. 

""" A cognitive constructivist problem-solving situation is likely to increase emotionality and jeopardize a 
student's relational attachments to teachers and to mathematics if not managed explicitly by the teacher 
(McLeod, 1992; Szetela, 1997; Windschitl, 2002). In such a situation where the student is expected to 
struggle over time with problems (with the teacher as guide or coach) she is likely to experience a range of 
emotions that includes frustration and anxiety. If the teacher helps her to expect this emotionality as a 
normal part of real problem-solving and to interpret and use it as a positive force in her process, her 
attachmoits to mathematics and mathematics teachers should strengthen, especially if the instructor 
provides herself as a reliable secure mathematics base whom the student can consult. 

'°°™" 1. The honesty and openness with which the pason is prepared for or informed of the loss or 

separation, is included in the mourning, and is allowed to mourn and to express her mourning over 
time; 
2 The quality of attachment to the mother (or attachment) figure before the loss or separation; and 
3. The continuity and quality of her relationship with other primary attachment figure/s after the loss or 
separation (Bowlby, 1980). 

"°™'' Marris (1974) has observed what he calls the "conservative impulse" universally at work in people's 
responses to loss, separation, and change. He proposes that this conservative impulse is based on the fact 
that people develop meaning and purpose in the context of cumulative and long developed attachments in 
relationships and circumstances. He notes that the cognitive process of assimilation of new understandings 
into a person's existing cognitive schema, observed by Piaget, is similarly conservative. Changes that cause 
disruptions to these attachments and that do not allow a person's engagement in the struggle to develop 
new purpose and forge new attachments or assimilate the changes into former attachments, are likely to be 
met with resistance and rejection, even when the changes can be seen by outsiders to be for the good 
(Ginsburg & Oppw, 1979; Marris, 1974; Piaget, 1967). 

""^ Reactions to separation or loss of attachment figure, or change impacting attachment bonds, have been 
found to follow a common bereavement process, beginning with 

1 . protest, involving confusion and searching for the lost object, sadness, yearning, anxiety, and 
anger towards the lost attachment figure or agent of change, then 

2. despair, depression, and disorganization, to 

3. detachment from the attachment figure as defense, and finally to 

4. acceptance of the loss (with ongoing sadness) if it is permanent, or to repaired attachment 
(typically accompanied by anger, distrust, and anxiety) if the attachment figure returns and resumes 
caregiving (Bowlby, 1980). 

"' A child's fransitional object was typically a physical object such as a soft blanket used to smooth the 
sometimes painfiil ttansition from complete dependence on her caretaker to her own autonomy (Winnicott, 
1989). I speculate that the emotional role of fingers, counters, and other raanipulatives or physical 
mathematical models, may be to fimction as transitional mathematical objects. These objects often smooth 
the transition from externally verifiable to internally known mathematical understandings and they may be 
comparable to a young child's fransitional object (or "blankie"). 



60 

CHAPTER III 
A NEW APPROACH: BRIEF RELATIONAL MATHEMATICS COUNSELING 

In chapter 1 I describe the problem of mathematics support center professionals 
not having what is needed to adequately help many typical college students to succeed. In 
chapter 2, 1 discuss the scholarship that led me to a hypothesis that relational counseling 
in conjunction with best-practice mathematics tutoring might address this problem. In this 
chapter, I show how I generated a counseling approach by adaptmg the theories I had 
studied to the realities of my practice as a learning center tutor. I describe how I used 
relational conflict psychoanalytic theory as a basis for understanding best-practice 
traditional mathematics support insights, as an approach to the student-tutor/counselor 
relationship, and as a remedy to difficulties standing in the way of student success in a 
learning support center context. I explain how the most important theoretical 
underpinning of my approach — relational counseling — can be applied in a mathematics 
academic support context to give a new way of looking at a student and at the tutor- 
student relationship. 

Drawing on my understanding of mathematics affective research findings and 
cognitive therapy I describe the development of tools designed to facilitate my 
understanding of students' affective and cognitive mathematics difficulties. I use the key 
terms from relational counseling that I redefmed in chapter 2 in the context of 
mathematics learning to show how relational counseling approaches may be used to 
elucidate how symptoms are related, their underlying causes, and possible treatments. 
Finally, I summarize ways mathematics tutoring and relational counseling can be 
integrated in practice by describing roles of key participants. 



61 

THE CHALLENGES OF ADAPTING RELATIONAL MATHEMATICS 
COUNSELING TO THE LEARNING CENTER CONTEXT 

The Therapy Approach and the Problem of Time Constraints 

How could a relational counseling approach be offered appropriately and effectively in 

the college setting? Practical consideration led me first to consider time and institutional 

constraints: For my purposes a major limitation of the relational conflict approach is the 

necessary long-term nature of the therapy. A typical college semester is usually 15 or 16 

weeks long. Realistically, potential contact time with a student is likely to be 

considerably less unless the student begins the semester conscious of his' need for 

assistance. Typically students recognize a need for support after the first quiz or exam 

which may be several weeks into the semester. I wondered if the short tkne available 

would be sufficient for a tutor to gain the in-depth understanding of the student that a 

relational approach promised. I was also concerned about the appropriateness of a 

therapeutic approach in an educational setting. 

Given the educational setting, counseling, with its problem-centered approach and 

counselor teaching/talking emphasis, seemed on the surface more appropriate than 

therapy, which has a person-centered approach and relatively long-term mvestigative 

emphasis based on close listening (Corsini, 1 995) . I considered the dual focus of 

relational mathematics counseling: mathematics tutoring and counseling. On the one 

hand mathematics tutoring is more problem-centered like counseling, since the focal 

problem is the student's understanding and ways of doing mathematics. Here the tutor is 

an expert in mathematics and takes on a coaching role as the student constructs new 

understandings from his already existing knowledge. On the other hand, relational 

conflict theory has generally been seen to involve client-centered therapy rather than 



62 

counseling. An approach to mathematics relationaUty then should be like therapy. It is 
person-centered, with the student considered more expert than the counselor in his own 
experiences, his personality, and his relationships. Here in contrast, the counselor's 
expertise is in investigating, listening, and interpreting how these explain the student's 
central mathematics relational conflict that needs resolution. While I call my developing 
approach relational mathematics counseling, it could perhaps be more accurately 
described as an integration of relational therapy into mathematics counseling. 

I was aware of brief therapies, but most were problem- rather than person- 
centered, like cognitive therapy, and I wanted an adaptation of relational therapy that was 
both problem- a«J person-centered. This adaptation exists in Stadter's (1996) brief object 
relations approach. Although some relational (psychodynamic) psychoanalysts resist 
shorter courses of therapy for all but narrowly specified problems (cf Sifiieos, 1987), 
brief therapy models such as Stadter's (1996) do apply relational (object relations) 
counseling to time-limited settings. Brief therapy incorporates cognitive counseling 
techniques and differentiates between the ongoing relational focus and the more 
immediate symptomatic focus. A brief relational mathematics therapy approach needs to 
incorporate the three relational dimensions, integrate pertinent CT/CBT approaches, and 
allow the immediate focus to be on the learning of mathematics. Such an overarching 
mathematics counseling framework could yield a nuanced understanding of students' 
mathematics mental health that could lead to treatment in the limited time available in 
college settings. It contains all the elements of an explanatory framework that can be used 
to understand and support mathematics cognition and affect in the context of students' 



63 

mathematics relationships. Such an approach could appropriately be offered through the 
academic support center. 

The Use and Misuse of Assessment Instruments 
As with best-practice traditional tutoring, the initial task in relational mathematics 
counseling is to understand the student, his understanding, and his approach to 
mathematics well enough to formulate an effective course of action. Understanding must 
be followed quickly by effective and flexible implementation of the course of action 
making constant adjustments in response to new insights and feedback from the student, 
the results of assessments, and effectiveness of approaches. A relational counseling 
approach differs from traditional practice, however, in how it changes the support 
professional's ways of looking at himself, at the student, and at their relationship during 
the tutoring process, as well as how it expands the scope of inquiry when investigating 
and intervening in the student's mathematics learning. 

A traditional approach to assessing or diagnosing a student's mathematics 
functioning is to use formal and informal paper-and-pencil assessments. These are 
generally used to identify the student's level on pertinent factors such as his mathematics 
affect and his aptitude, achievement, and/or developmental level on the mathematics to 
be attempted in the course. 

Cognitive assessments I had used included in-house mathematics placement 
instruments. Scholastic Aptitude Test (SAT) quantitative scores, and in-class tests and 
quizzes. To assess affective orientation and identify possible affective symptoms of 
mathematics difficulty I had previously used a number of diagnostic past-experience 
questions, mathematics affect, and orientation surveys that explored students' beliefs. 



64 

attitudes and feelings. In researching for this study, I became familiar with other 
instruments. My first inclination on lighting upon relational conflict theory as my 
framework was to abandon these instruments and surveys, principally because of my 
frustration with not knowing how to prioritize, understand, and use the data they 
gathered. I quickly realized however, that given the short time available in a semester 
worked against the relatively time-consuming relational therapy approach to data- 
gathering so efficient data-gathering instruments would be necessary. Importantly, I 
realized that the relational conflict framework was my key to prioritizing, understanding, 
and using the data gathered by these instruments: Far from abandoning them, it seemed 
that my new approach required their use. 

I looked for assessments to help students become conscious of their present 
condition with respect to their mathematics learning, both affective and cognitive, and 
become aware of what that revealed about their established relational patterns. I was 
aware that a new approach might deeply challenge not only the traditional uses of 
assessments but also students' conceptions of what assessments could and could not say 
about them. 

Stephen J. Gould (1981) writes: 

Few tragedies can be more extensive than the stunting of life, few injustices 
deeper than the denial of an opportunity to strive or even to hope, by a limit 
imposed from without, but falsely identified as lying within, (pp. 28-29) 
Here Gould refers to the historical use of psychometric "biological labeling" to define 
and limit the intelligence or abilities of groups or individuals in the U.S. I expected that 
in introductory college mathematics-related classes in the U.S., there would be students 



65 

who had been subjected formally" or informally'" to such a denial of opportunity based in 

a limit imposed by inappropriate interpretation of testing results in mathematics. I had 

seen the effects of this denial to be affective, cognitive, and also relational. It negatively 

impacted a student's overall mathematics fimctioning, that is, his mathematics mental 

health. Accordingly I looked for assessment tools that could help the student become 

consciously aware of his mathematics limits, of his beliefs about those limits, and of his 

attitudes, emotions, and relationships related to his Umits so that we could explore and 

detoxify the source of deceptive limits and constructively deal with real ones. I 

determined that in any use I made of assessments I would keep central the possibility — 

indeed, the expectation — of changes over time in the assessments for each student. 

WHAT THE RELATIONAL MATHEMATICS COUNSELOR NEEDS 
TO KNOW ABOUT THE STUDENT 

Three avenues of inquiry emerged as important when I considered what 
information I needed early in the process to begin to understand a student's mathematics 
relational patterns and provide a way of discussing those with him. First, I wanted to 
capture the student's sense of where he had come from mathematically, where he was 
now, and what he thought were his key issues. Second, I needed to know how the student 
was actually dealing with this mathematics class, the course instructor, and the content, 
both in the classroom and out. Third, I hoped to fmd ways to see myself in relationship 
with the student, and him in relationship with me to inform my interpretation of the &st 
two. 

Determining How the Student Sees Himself 

In order to explore the student's sense of his mathematical progression, his 
current placement, and what he considers his key mathematical issues, a two-pronged 



66 

approach seemed feasible: (a) First, during counseling sessions, I would use direct and 
indirect questioning to analyze his mathematical orientation, approaches, and 
background, and (b) second, outside of the counseling session (e.g., in class, for 
homework), I would use strategic self-report surveying of factors I considered pertinent 
to a student's mathematics relationality, such as his beliefs, attitudes, and feelings around 
mathematics that I thought might be difficult to systematically gauge during counseling 
sessions. I could use his survey responses in counseling as a vehicle to focus on issues 
that might not otherwise arise. 
During the Counseling Session 

Mathematics background and experiences. In mathematics relational counseling, 
taking personal history that focuses on the person's experiences with significant 
mathematical others is likely the first essential to establishing a suitable relational focus 
and to a proper understanding of how to deal with his particular mathematics learning 
needs (for relational therapists' use of history taking, cf Luborsky & Luborsky, 1995; 
Stadter, 1996). In mathematics anxiety reduction clinics (cf Tobias, 1991), in some 
academic support settings, and in research studies (cf Mau, 1995), it has been standard 
practice to invite adult students to tell or write their mathematics learning histories or 
autobiographies to explore their present negative affect in the mathematics learning 
situation, but is rare in the context of a college mathematics course, either in class or in 
tutoring because of time pressures to focus on course content. Since my new approach 
required it, however, I developed a Mathematics History Interview Protocol (see 
Appendix A) based on findings from a qualitative research study I conducted into college 
students' mathematics identity development and from my subsequent mathematics 



67 

academic support work with college students (Knowles, 1 998). Important areas of 
inquiry include not only relational experiences with teachers, parents, peers, and others, 
but also which completed high school and college mathematics and areas of self- 
perceived mathematics competence and incompetence. I expected that because of time 
pressure and the urgency of the current mathematics course focus, this history will 
probably need to be gathered over several sessions, and history gathering would need to 
be integrated into the ongoing mathematics tutoring process so that students can see its 
connection and relevance to their current mathematics objectives. 

Metaphor. "Metaphors are concrete images that require us to fmd the threads of 
continuity and congruence between the metaphor and the primary subject" (Deshler, 
1990). The primary foci for students in mathematics counseling should be mathematics 
and themselves as mathematics learners. In my previous practice, I had asked students 
write metaphors for their experience of mathematics but I had not known how to explore 
beyond the obvious "threads of continuity and congruence" with students' mathematics 
learning such as personal affective orientation to mathematics or beliefs about what 
mathematics is. Now I realized that metaphor might also give students access to their 
underlying relationships within their mathematics learning in an open-ended, indirect, 
imaginative way. The relational perspective gave me a way to explore a metaphor with a 
student, noticing clues to his sense of mathematics self, his internalized mathematics 
presences, and his mathematics or mathematics teacher attachments. 1 could see how a 
student's metaphor might provide a unifier or common thread to piece together other data 
to understand the student's central relational conflict patterns. In counseling, I needed to 
express my assumptions about his intended meaning in order to have the student clarify 



68 

or amend my perception. Such joint exploration seemed likely to unearth underlying and 
possibly unconscious relational connections. During and at the end of the course of 
counseling, students could reconsider their initial metaphors to see whether and how they 
had changed and what, if any, changes might signify with regard to outcomes of the 
course of counseling. 

Mathematics negativity. In the mathematics learning situation, students with 
negative beliefs about their mathematics world, their mathematics selves, and their 
mathematics futures'^ tend to exhibit symptoms more like those of situational depression 
than the more commonly assumed anxiety. This mathematics "depression" can be 
debilitating in the learnmg situation, and students thus afflicted seem quite likely to give 
up quickly, withdraw, or fail. The severity of the negative outlook may change from 
week to week and, with that, the student's energy to struggle with the coursework, in an 
inverse relationship between energy and severity of negative outlook. Dweck (1982), 
Beck (1977), and others have found that having a person articulate her negative self- 
statements may be the first step recognizing their irrationality and changing them. Having 
clients respond to questions about their world, themselves, and their future each time they 
met vnth a counselor has been found to help them and the counselor tackle negative self- 
statements in an ongomg and timely manner (Beck, 1976, 1977). Therapists using this 
method were also able to gauge the severity of the cUent's negativity/depression and 
sometimes to prevent him from harming himself (Al-Musawi, 2001; Simon, 2002; 
Sprinkle et al., 2002). Analogous to this self harming m the mathematics learning 
situation is a student's sabotaging his chances of success by avoiding work or even the 
mathematics class when his negativity and hopelessness become overwhelming. 



69 

In order to help my students become conscious of their thinking so they could 
consciously deal with it rather than withdraw, for this study I developed a set of line 
scales, each of which allows a range from positive to negative responses about the 
student's current mathematics course, self, and future that week {JMK Mathematics 
Affect Scales, see Appendix A). At each session the student will fill in the scales, and we 
could compare his responses to previous ones and discuss changes in relation to external 
circumstances, his progress with the course, and thoughts about himself We might look 
for connections with his relational challenges and use this as feedback to help clarify the 
focal relational conflict he is working to resolve. We might discuss changes in routines 
and in his thoughts about hunself that he might try to implement over the following week 
in response to the current evidence. 
Outside the Counseling Session 

The traditional means for finding out how a student sees himself as a mathematics 
learner has been the self-report affect survey. Surveys requiring responses on a five- or 
seven-point Lickert scale can be administered quickly in class or as a homework 
assignment. I wondered if I might collect such data on affect that could help provide a 
fiiller picture of the student that could be missed if I relied solely on conversation m the 
counseling session. 

Researchers have found two major areas of affect that interact directly with 
mathematics cognition (albeit in complex and not always explicable ways) (cf McLeod, 
1992). They are mathematics feelings (specifically, anxiety) and mathematics 
beliefs/attitudes (and attributions based on these beliefs/attitudes). I determined that I 
needed to find ways of observing or measuring students' levels of anxiety and 



70 

helplessness in new learning and testing situations (possibly indicating a damaged or 
underdeveloped mathematics self) as well as curiosity and mastery orientation (possibly 
indicating a healthy mathematics self) (Carter & Yackel, 1989; Skemp, 1987). 1 thought 
a survey of each of these two areas — feelings and beliefs — that investigated key factors 
linked with mathematics understanding and achievement might provide important points 
of discussion and clarification in counseling. In addition, if used as a pre- and posttest, it 
seemed possible that such surveys might reveal movement or change over the course of 
counseling. I weighed the limitations of such surveys (e.g., closed questioning, 
insensitivity to precision or depth or range of actual student feelings or beliefs) against 
their benefits (e.g., quick assessment [using small constellations of items] of research- 
confirmed key factors, and links with a student's underlying and overt relational patterns) 
to assess what and whether surveys of affective issues could be helpful in the counseling 
process. 

Finding Out How the Student Does Mathematics Now 

Mathematics diagnostics. Because I conjectured that students' mathematics 
relational challenges (especially their sense of mathematics self) might be closely linked 
with poor attachment to mathematics, I looked for diagnostics that could be administered 
in class or during a counseling session that could discern between perception and reality 
and thai were linked closely enough with current course content to be useful guides to 
appropriate relational conversation. 

Whatever the emphasis of an introductory college mathematics course, 
arithmetical prowess in number (small and large) and operation sense and the student's 
understanding of the algebraic variable seem to be pivotal areas to be explored. I 



71 

surmised that strategic use of an arithmetic and/or an algebra diagnostic could help both 
student and counselor better understand the affective and cognitive impact of the 
student's mathematics learning history. Once we had that information the student and 
tutor could jointly plan strategic mathematics interventions for this course (see Appendix 
B for the assessments I devised or adopted : Arithmetic for Statistics Assessment, 
Knowles, 2000; the Algebra Test, Sokolowski's,1997, adaptation of Brown, Hart and 
Kuchemann's,1985, Chelsea Diagnostic Algebra Test) 

If the course had a specific applied emphasis (e.g., statistics) I wondered if a 
specific diagnostic of that application could also be helpful. For an example see 
Appendix B for Garfield's Statistical Reasoning Assessment used in my pilot study. 

Mathematics course achievement. Mathematical tasks required in the course are 
naturally central in coimseling. Students react to the grades they receive on course 
assessments — exams, homework assignments, projects — differently, I surmise, because 
of differences in their background experiences and relational challenges, and they also 
react differently to these grades. A pivotal challenge in counseling is to analyze a 
student's products with him in a way that helps him interpret his grades constructively. 
The counselor must try to understand his reactions and to help modify them if necessary 
in order that the student will approach the next assessment with a sense of responsibility 
and with a developing sense of his mathematics self In this testing situation, the student 
feels most acutely that his mathematics self is being judged. He may evidence conflicting 
motivations and behaviors (e.g., wanting to succeed but also wanting to protect a 
vulnerable sense of mathematics self by not trying, so as to avoid judgment of ability). 



72 

These heightened conflicts are likely to become clearest during exam analysis dicussions, 
so these discussions create special opportunities for relational counseling. 

Mathematics practices and behaviors. How the student actually does mathematics 
may differ from how he perceives himself doing it. He may do it diflsrently in different 
settings, and the counselor's observations and exploration of discrepancies should make 
the student aware of approaches that he may need to modify. The settings where the 
student does mathematics mclude the classroom, his home or dorm, and the learning 
support center. Typically the mathematics counselor can observe the student directly only 
in the learning support center, although he may be able to arrange classroom visits and/or 
receive instructor observation reports (with student permission). It would seem however 
that counselor observation of the student doing mathematics in different settings, 
particularly in the classroom could be crucial for a clear understanding of the student's 
mathematics relational issues. 

The Student-Counselor Relationship 

As I envisaged relational mathematics counseling, I realized that my relationship 
with the student and his with me could be vital to understanding his core challenges, but 
only if I purposely made our relationship a central object of inquiry and even, at times, a 
topic of discussion. I noted the pivotal place relational therapy gives to the client's 
transference of past analogous relationships to his relationship with the counselor and the 
counselor's countertransference responses to the client, acknowledging that much of the 
client's relationality is discernable through understanding and interpretation of this 
mterchange. When I considered how I might integrate this observation and analysis of 
our relationship into what the student understands to be essentially a mathematics tutor- 



73 

tutee relationship, I realized that if 1 self-disclosed when I became aware of my own 
countertransferential impulses and asked about the student's sense of what was going on 
and who they thought I should be and what they though I should be doing when I became 
aware of being other than who I was, we might establish a place for exploring what it 
might signify about their mathematics relationality. This approach seemed appropriate in 
the learning support setting, but I was aware making countertransference and transference 
issues explicit and be explored would likely differ markedly from student to student. 
However, I could now admit my own countertransference and my experience of the 
student's transference as data regardless of whether explicit discussion with the student 
felt appropriate. As I considered the unportance of transference to a relational 
mathematics counseling approach, it also became clear to me that I needed to arrange 
supervision meetings in order to review and assess my transference-countertransference 
interpretations with a person knowledgeable in counseling psychology. 

Understanding the Student's Mathematics Mental Health Conditions 

My interpretation of a relational view of mathematics mental health holds that a 
student's relational patterns are adaptive. That is, he has developed ways of relating to 
mathematics, instructors, and required mathematics courses that serve his sense of 
mathematics self. His adaptations to mathematical circumstances may be conducive to 
growth and positive development; they may be detrimental and skew or stunt his 
development; or they may be somewhere in between. A student's state of mathematics 
mental health may range from sound to poor, depending on the sense of mathematics self 
he is attempting to maintain and the extent of conflict between contradictory goals he is 
experiencing. 



74 

I had noted certain conditions or sets of indicators (or a syndrome) that could be 
used to describe a student's state of mathematics mental health. These conditions could 
be manifested as cognitive, as affective, or both. I believe that these may be best 
understood in the context of a student's mathematics relationality. Understanding these 
conditions or sets of indicators seemed key to helping a student focus quickly on his core 
relational challenges. 

Mathematics Cognitive Conditions and Relational Counseling 
Research and experience have informed me that the cognitive conditions most 
likely to negatively impact college students' achievement are: (a) a procedural approach 
to mathematics learning, (b) the lack of a "profound understanding of fundamental 
arithmetic" (L. Ma, 1 999) primarily number and operation sense, (c) weak connections 
between arithmetic and algebra; (d) underdeveloped understanding of the algebraic 
variable, (e) poor or counterproductive problem-solving strategies and monitoring and 
control skills, (f) poor course management skills or (g) any combination of these. As a 
mathematics counselor I would have to not only assess a student's cognitive standmg, 
considering these categories, but also consider their impact on the development of his 
mathematics self and his relational patterns. I would then have to prioritize tutoring 
attention his cognitive conditions in relation to the demands of the course and his 
limitations. 

Cognitive Conditions Related to Personal Cognitive and 
Environmental Attributes 

Students with strong particular learning style^ inclinations may display learning 
strengths or weaknesses depending on the particular learning environment.^' If there has 



75 

been a long-term mismatch between a student's learning style and mathematical learning 
environments, unless he has been able to be flexible, he may have experienced less 
success than his potential would indicate, along with an associated loss of confidence in 
his ability. 

Although there is powerful evidence that average children can learn mathematics, 
many, and especially (but not exclusively) those from disempowered groups, are in 
classrooms where their ability is judged inaccurately. They are often judged to be lower- 
ability than they truly are and, perhaps worse, the ability they are considered to have is 
judged as fixed (Downs, Matthew, & McKinney, 1994; Sadker & Sadker, 1994; Secada, 
1992). Most U.S. students have experienced formal or informal tracking into ability 
groups since the early elementary grades. Likewise students with diagnosed leammg 
disabilities, ahhough cognitively capable, are likely to have been subjected to even lower 
teacher expectations. Piaget (1973) goes further than Krutetskii (1976) in rejecting the 
notion that some people have a math mind and many do not, but most U.S. college 
students have entrenched beliefs about their own math ability that have restricted the 
development of their ability and led to learning gaps. They may have been put in lower 
tracks and given less coursework in high school, and they may have taken fewer courses 
thus jeopardizing their achievement in college (cf Sells, 1976; Schoenfeld, 1992). 

There are complex relationships among students' race, language, ethnicity, SES, 
and gender, and their mathematics achievement (Secada, 1992). There is no credible 
evidence that any of these factors or combination of factors affect potential to succeed. 
There is, however, consistent evidence that schools' differential fmancial resources, 
school cultures, and teacher race and ethnicity, attitudes, and expectations negatively 



76 

affect persistence in mathematics course-taking, achievement, and especially the 
academic confidence of students from disempowered groups. Students from a 
disempowered minority group who have been schooled in a majority setting where 
teachers who are predominantly of the dominant culture is likely to experience minimal 
respect for his own cultural norms or for the non-English language he speaks. Should this 
be true, he has likely experienced minimal mirroring from the teacher and insufficient 
support for his budding mathematics self 

The development of a student's mathematics self, is affected by myriad personal 
and environmental factors and their interactions. Students with underdeveloped or 
damaged mathematics selves tend to blame their difficuhies on their own (imagined) 
intrinsic inability or some other defect because they have been treated as if they are 
inadequate. It has been relatively rare that a teacher is aware of and takes responsibility 
for his part in his student's difficulties in learning mathematics. A relational approach to 
students who have suffered such assaults on their mathematics selves should involve 
carefial attention to what they can do mathematically, building on their abilities and 
understandings using methods compatible with their learning styles, and refiitmg their 
"no math ability" theories with evidence of their own work and thinking. In other words, 
they need teacher-mirroring and support of their vuhierable and undeveloped 
mathematics selves. 

Mathematics Pedagogy and Cognitive Conditions 

The mathematics self seems to be the most central dimension in the development 
of healthy, flexible mathematics relational patterns. The principal means for this healthy 
development is good mathematics teaching"" in an enviroimient where the student's 



77 

mathematics self is accepted, coached, and challenged. When that has not occurred or has 
occurred intermittently, cognitive symptoms emerge, such as rigid reliance on memorized 
steps or difficulty in adapting to slightly different wording or appearance that are 
observable in the adult student's arithmetic, algebra, and problem-solving work in class, 
on exams, and in the counseling session. How these cognitive symptoms interact with 
students' affective symptoms and what they tell about the student's overall state of 
mathematics mental health is investigated in this study. We can expect arithmetical 
weaknesses and imcertainties to have deeper, more longitudinal and negative implications 
to the mathematics self (identity) than algebraic weaknesses (if arithmetic is intact). 
Number and operation sense weaknesses may be especially toxic, depending on their 
severity and pervasiveness. As an example, Janet's lack of automatic access to her 
multiplication and addition facts (see chapters 1 and 2) slowed her progress in 
precalculus and seriously undermined her confidence. However, algebraic weaknesses 
will invariably also strongly impact present functioning negatively. How cognitive 
symptoms specifically affect an individual's present ability to learn new mathematical 
content will be a function of a combination of the course difficulty, the way it is taught 
relative to the student's needs, the relational and mathematics climate of the classroom, 
the extra support available, and the way the student's mathematics relational patterns 
interact with these factors. Vuhierable students may include not only those with cognitive 
preparation deficits but also some whose cognitive preparation is adequate but who are 
nevertheless not confident for other reasons. 

Researchers such as Skemp (1987) and Buxton (1991) have shown links between 
affective and cognitive symptoms that have their source in poor mathematics pedagogy. 



78 

In particular, predominantly procedural teaching with the teacher as the sole authority on 
the mathematics leaves the student vulnerable to helplessness and anxiety because he has 
recourse only to memory or the teacher's logic rather than to the connections he could 
make himself if he has learned and understood it conceptually. 

Cognitive Conditions and Relationality 

Attachment to mathematics. Few elementary teachers have what Liping Ma 
(1999) calls the "profound understanding of fundamental arithmetic" required to 
understand the problems, and few are able to translate their understanding into practical 
activities for their students. Thus they have to teach their students procedures rather than 
concepts. These students tend to develop a narrow procedural knowledge of arithmetic 
that links poorly with algebra because of the need to generalize beyond procedure to a 
more abstract statement of relationship. Students' knowledge of and beliefs about 
mathematics and about themselves doing mathematics may be distorted. If they have not 
developed a secure attachment to mathematics that can enable them to be flexible and 
venture into new learning this distortion may be extreme. 

EMOTIONAL CONDITIONS AND RELATIONAL COUNSELING 

Anxiety 

Much of the negative affect that students experience while doing mathematics has 
been lumped under the label "mathematics anxiety." Educational research supports a 
relationship between mathematics anxiety and poor performance although that 
relationship is not unequivocal nor is the effect always significant when it occurs 
(Hembree, 1990; McLeod, 1992). According to the Yerkes-Dodson (1908) principle 
(performance related to arousal roughly by an inverted U), students who experience 



79 

moderate levels of arousal (whether they interpret that as positive or negative) will do 
better on a test than those who experience either too little or too much arousal. What 
exactly mathematics anxiety is and what its causes are have been the matter of much 
debate and many studies and factor analyses (Ma, 1999; McLeod, 1992). Part of the 
difficulty is that its etiology, triggers, and expression differ from person to person. A 
relational counseling approach, I believed, would provide the mandate and opportunity to 
enable students to reveal and explore these individual differences. But because of 
semester-long limitations, I wanted an instrument that would differentiate some factors in 
mathematics anxiety and provide a starting point for discussion with individuals in the 
counseling situation. 

Analysis"" of the literature of attempts to define and measure mathematics and 
testing anxiety have found a number of dimensions that affect students' performance in 
sometimes singly, sometimes in combination, and always in relation to other dimensions 
all in varying degrees. The pertinent dimensions are often agreed to be: (a) the 
mathematical situations that engender anxiety (e.g., every day life vs. classroom; within 
the classroom: testing vs. class work versus homework); (b) the type of mathematics 
involved (e.g., arithmetic vs. algebra); (c) the cognitive precursors to anxiety (e.g., poor 
exam preparation); (d) whether the mathematics activity is solitary, with peers, or public; 
(e) to what extent the student suffers from strong chronic anxiety or experiences anxiety 
easily (trait anxiety); (f) the type and intensity of anxiety engendered by the situation 
(state anxiety, cognitive worry); and (g) the immediate and long-term physical, affective, 
and cognitive effects of the anxiety. 



80 

I determined that if I understood a student's mathematics affective history and its 
effects on the different dimensions of his mathematics relationships and, flirther, if I 
observed and experienced his resultant relational patterns, I might be able to 
contextualize his anxiety. I searched among the many formal and informal instruments 
for one that surveys affective response to mathematics cognitive and situational factors. 
This seemed particularly urgent because of my perception of the centrality of 
mathematics cognition in the development of the mathematics self. I chose Ferguson's 
(1986) Phoebus (which I renamed as My Mathematics Feelings survey see Appendix B 
and see endnote ix) to be used in conjunction with the student's and my observation and 
discussion of his testing behavior. Other pertinent factors would emerge during 
counseling and their relational etiology also could be explored. 

I would first consider normal anxiety that is engendered by a dangerous situation, 
before looking for a psychological cause originating from a disturbance of mathematics 
self, internalized presences, or interpersonal attachments (Bowlby, 1973; Fairbairn, 1992; 
Freud, 1 926; Kohut, 1 977). In this context such causes as inappropriate placement in the 
class (indicative of prerequisite knowledge gaps), insufficient strategic preparation for an 
exam, or poor problem-solving, monitoring and control skills would genuinely endanger 
the student's chance of doing an exam successfully. These examples constitute 
appropriate causes of normal anxiety. 

Once such normal anxiety has been ruled out, I would consider the relational roots 
of a student's anxiety. 



81 

The Mathematics Self and Anxiety 

Anxiety related to assaults on the development of self is what the founder of self 
psychology, Heinz Kohut (1977) describes as disintegration anxiety, "an ill-defined but 
intense and pervasive anxiety accompan[ying a sense that the] self is disintegrating 
(severe fragmentation, serious loss of initiative, profound loss of self-esteem, sense of 
utter meaninglessness)" (p. 103). I have seen this when a student with a deep sense of 
his own inability to do mathematics becomes inarticulate and paralyzed when called on in 
class or experiences panic, mental disorganization, helplessness, even physical pain when 
taking a test. Could he be experiencing a form of the disintegration anxiety Kohut spoke 
of? Is this part of himself so malformed or underdeveloped that when his mathematics 
self is being scrutinized by a public question or a test, especially in mathematics class, he 
feels his self disintegrating to the extent that it might even threaten the rest of his 
developing academic self (cf Lenore in Fiore, 1999; Tobias, 1993)? 

I envisaged that counseling help for a student suffering so could take a two- 
pronged approach. The counselor could help the student to connect with mathematics, to 
recognize and own his developing understanding, and to expand his tolerance of the 
anxiety engendered by not knowing or understanding it all immediately; At the same 
time, the counselor, student, and instructor might explore alternate alternative 
arrangements in class work or testing designed to alleviate anxiety. For example, the 
instructor could signal that the student will be the next person to be asked an identified 
question so he has time to prepare an answer, or exam questions could be given one at a 
time. 



82 

Internalized Presences and Anxiety 
When a student has developed and repressed bad internalized presences in 
response to unsafe and abusive mathematics learning situations, or has established 
mathematics as a punitive internal saboteur or superego, these internalized presences may 
cause him to worry that his mathematical products are bad or wrong even when they are 
not. He may have internalized his frightening third grade teacher who made him stand at 
the board for long hours humiliated and unable to do the required problem and this 
teacher's influence may be manifested during the college exam, insisting that he still 
cannot do it and recreating the mind-numbing anxiety he experienced back then (cf Terry 
in Fiore, 1999). During the exam, he may have to contend with the anxiety engendered by 
the prospect of his exam grade pronouncing judgment on his worth as a person (cf 
Buxton, 1991). 

Interpersonal attachment and separation anxiety. Involuntary separation from a 
person's attachment figure often causes distress and creates disturbance in that 
relationship when the attachment figure returns, no matter how short the separation or 
how well the separation was managed. If the person subsequently comes to believe there 
is risk of fiarther separation he is likely to become acutely anxious (Bowlby, 1973). A 
student may experience such acute anxiety if he has done well in mathematics and 
enjoyed positive relationships with teachers but has been separated from these good 
experiences and subsequently had a bad experience. He may have done badly in a course, 
clashed with or been ignored by a teacher. Separation anxiety is a natural response in 
children and adults'^ whose access to their attachment base is denied or threatened or 
whose attachment figure is unresponsive. Maladaptive responses to separation, loss, or 



83 

change can be an apparent lack of response (i.e., detachment) or an mtense response (i.e., 
extreme anxiety or phobia) (Bowlby, 1973; Freud, 1926). 

In a study of instructor-caused onsets of students' mathematics anxiety, Jackson 
and Leffmgwell (1999) found that responses that could be classified as separation anxiety 
arose from the perceived inaccessibility or lack of responsiveness of the mathematics 
caregiver, the instructor." Experiencing inaccessibility or lack of responsiveness from 
previous teachers can negatively affect students' responses to their current teacher's 
offers of help as a secure base. Without understanding and intervention this separation 
anxiety may persist. 

Students who have once experienced success in mathematics but have 
subsequently suffer a loss of competence because of poor teaching, course placement, or 
other external events may experience separation anxiety in relation to the mathematics 
itself They may be newly uncertain of its accessibility and reliability. Without 
counseling mterventions to reconnect them to their once-secure base in mathematics and 
their sound ability to negotiate the current course, this separation anxiety may cause them 
to fail or do poorly in mathematics courses they are capable of mastering. 

This exploration of the relational origins of mathematics anxieties led me to see 
that once the student and I had determined through the My Mathematics Feelings survey 
and conversation that his mathematics anxieties existed and were troublesome, we could 
go further and distinguish their origin in different relational dimensions and devise 
targeted interventions that could look quite different depending on the dimension of 
origin. 



84 

Learned Helplessness and Depression 

Anxiety is not the only emotional response to mathematics stress. In my 
experience, students who suffer from mathematics negativity (see above) expressed as 
learned helplessness or even depression with or without anxiety are just as prevalent. 
Learned helplessness has been linked with both situational and clinical depression 
(Seligman, 1975). Dweck and Reppucci (1973) found that a student may come to believe 
he is helpless under one set of circumstances but not under others. This supports 
Seligman's (1975) notion of situational learned helplessness or depression. It may not be 
so much the mathematics itself but the way it has been taught that renders students so 
vulnerable to learned helplessness in its face (Boaler, 1 997; Carter & Yackel, 1 989; 
Dweck & Reppucci, 1973; Piaget, 1973; Skemp, 1987) Mastery-oriented, positive 
students may exhibit helplessness and depression- like symptoms in certain mathematical 
contexts. Learned helpless and depressed people believe that the situation they are in is 
beyond their control; there is little or nothing they can do to change the outcome. 

It is not unusual to fmd one or two students in any class of 30 who view their 
mathematical past, present, and iuture with despair. A mathematically depressed student 
sees himself as mathematically deficient; he considers the present mathematical demands 
excessive; and he views his fiiture as impossible. He may want to drop the course he is in 
now and he will seek any alternative to the looming mathematics course to follow. 

A depressed person's negative orientation and behavior influence other people 
whose responses in turn influence the individual (Bandura, 1 977). For example, 
emotional withdrawal may elicit rejection or criticism that in turn aggravates the patient's 
negative self-cognition and thus his depression. A mathematically depressed student may 



85 

avoid classes, homework, or the learning support center. This avoidance behavior inay be 
interpreted as laziness or irresponsibility and result in censure rather than sympathy. 
Alternately, a mathematically depressed student may become excessively dependent on 
the mathematics counselor or the instructor and seem unable to proceed on his own. 
Student Beliefs, Helplessness/Depression, and Mathematics Pedagogy 

Students develop beliefs about mathematics and their ability to understand it that 
are closely linked with the beliefs and practices of their teachers and the effects on their 
mathematics orientation and self concept. In the U.S., the most detrimental belief about 
mathematics and mathematics learning that has the most far-reaching negative 
consequences for students is: "Learning mathematics requires special ability, which most 
students do not have" (Mathematical Sciences Education Board, 1 989, p. 1 0)."' The belief 
that ability is a trait rather than a malleable quality has been linked to learned 
helplessness in mathematics learning situations (Dweck & Wortman, 1982). It amounts to 
a type of mathematics gene theory that is applied in both a positive and negative manner. 
A student who identifies with a family member who is "good at mathematics" is likely to 
believe he also has the potential to be "good at mathematics," but students who identify 
with a family member or members who "could not do mathematics either" are more 
common and are likely to find this belief debilitating. It has been found that a student's 
beliefs about his achievement lead his to attribute outcomes to one of two central causes: 
his ability or his effort. Thus a student who believes his ability is low and unchangeable 
is likely to attribute a poor score on an exam to his (poor) ability. If a student attributes 
both his failure to lack of effort and also success to his (soimd) ability, he is ascribing to 
beliefs that generally underlie a healthy mastery approach. On the other hand these 



86 

attributions may instead be an all-powerftil, all-knowing (grandiose) mask for an 
underlying fear that one might not be able to do it — and that one has no intention of 
trying because of the risk of being found out (see below). 
Student Beliefs, Achievement Motivation, and Helplessness/Depression 

Achieving a high grade or some other recognition, also termed performance 
achievement motivation, often becomes more important and more possible than learning 
with understanding in the compulsory and competitive U.S. school systems. Piaget 
( 1 967) sees learning achievement motivation to be related to two important factors: (a) 
the "moderate novelty" of the new task, and (b) reasonable proximity and accessibility of 
learning, given levels of prior understanding. An mdividual's curiosity is aroused by the 
"moderate novelty" of an object in relation to his prior experience; this curiosity 
motivates his to investigate, learn, and achieve understandmg. The students in this study 
brought many different motivations to their tasks of succeeding in a mathematics course. 
These stem from their prior experiences, are related to their present ambitions, and affect 
how they would do in the course. 

Both learning and performance achievement motivational patterns have been 
found to be affected by students' sense of worth (Dweck, 1986). A sense of contingent 
self worth and a belief that their intelligence is fixed typically lead students to make 
performance as their primary goal in learning; they work only to be seen and judged to be 
successful. They will not approach a task with confidence (mastery orientation) unless 
they perceive their ability to be high for that task. If they perceive their ability to be low 
they are likely to become discouraged and even helpless. If they have a choice of learning 
tasks, some tend to choose tasks that are below their ability in order to ensure good 



87 

performance or they will choose tasks well beyond them that no one would expect them 
to complete successfully. 

Students who believe their intelligence is malleable show more adaptive 
motivational patterns; they typically make learning their primary goal. These students 
typically approach tasks with a mastery orientation regardless of whether they perceive 
their ability to be low or high in relation to the task. They choose learning tasks because 
the tasks are personally challenging rather than first considering whether they are able to 
do well at them (Dweck, 1986). These students are likely to be discouraged and anxious 
and become helpless in fast-paced, text-based, procedural classroom where they find 
learning and understanding the mathematics difficult or impossible (Boaler, 1997). The 
optimal conditions for learning achievement motivation to lead to understanding and not 
be frustrated include these principles: 

1 . Students need to be encouraged to make it their personal goal to solve the 
problem; the tasks themselves need to be "appropriately problematic" (Hiebert, 
et al., p. 51); 

2. The culture of the classroom must be a secure base that supports and allows 
time for struggle, reflection, and communication; 

3. Students need to see ways to use the tools they already possess to begin the task. 
(Hiebert et al., 1997) 

Apparently similar classroom behaviors may stem from quite difl^erent 
motivational orientations linked not only to the student's sense of mathematics self but 
also to the mathematical tasks and learning environment. 



88 

Student Beliefs and Helplessness/Depression: Developing 
a Survey Instrument 

Students' beliefs about their mathematics selves, world, and future have been 
researched extensively and the links between these beliefs and their mathematics course 
persistence, behaviors, and achievement have been thoughtfully studied. As noted above I 
had developed the JMK Mathematics Affect Scales (see Appendix A) to quickly gauge 
students' immediate operating beliefs on a session-by-session basis. I wondered if in 
addition I could develop or find and adapt an instrument that would survey underlying 
factors researchers had linked to mathematics negativity or helplessness. 

Whereas a mathematics anxiety instrument is intended to assess students' short- 
term emotional responses, a belief survey looks more at stable long-term underlying 
beliefs and attitudes. These may help to explain the student's short-term emotional 
responses as well as established mathematical behaviors. I looked for a self-report survey 
instrument around beliefs about self and mathematics that included statements about the 
following: 

1 . Mathematics as procedural or conceptual; 

2. Mathematics self as learned helpless through mastery oriented in mathematics 
learning situations; 

3. Links between mathematics beliefs and mathematics self beliefs; 

4. Achievement motivation: performance through learning motivation 

5. Personal characteristics and societal myths: Fennema and Sherman (1976), 
Fennema (1977), Kogelman and Warren (1978), Tobias (1993) and others 
have shown links between these and mathematics anxiety and debilitation of 
mathematics achievement. 



89 

I did not include locus of control as a factor, although 1 knew that a student's 
locus of control (whether he sees himself or some external entity such as luck or the 
teacher, as the controller of his outcomes) has been found to be an important factor in his 
mathematics achievement (cf Nolting, 1990). I preferred to gauge this and also a 
student's locus of responsibility (whether he sees himself or others as responsible for 
what happens to him) directly from cues in the counseling setting. 

With some modifications, Ema Yackel's (1984) Mathematics Beliefs Systems 
Survey with some modifications fit my criteria. Its chief attraction for my purposes was 
the careful investigation it provides of procedural versus conceptual beliefs about 
mathematics, based on Skemp's (1987) analysis (see Appendix B). I reasoned that 
analyzing clusters of items with the student could help him become conscious of beliefs, 
attitudes, and conditions whose relational origins we could explore and that he would be 
free to modify. I then considered what that exploration of relational origins of negative 
and counterproductive beliefs might reveal. I found the relational dimensions of a) the 
student's mathematics self and b) his interpersonal attachments to be particularly 
vuhierable to development of different types of depression. Because of the difference 
between the origins and hence potential remedies for these depressions I needed to clarify 
how to distinguish them when their initial presentation was likely to be similar. 
The Mathematics Self and Depression 

While CT describes the manifestations of mathematics depression, relational 
theory traces the origins of the depression and points to relational remedies. If a student's 
mathematics self has been under-stimulated because of chronic teacher neglect, his 
mathematics self will likely be underdeveloped. When such a student is faced with a 



90 

mathematics challenge, he is likely to experience a vague but pervasive sense of 
depression and excessively low mathematics self-esteem. His depression will feel like 
emptiness, a sense of not really being alive mathematically. He may believe his 
mathematical self does not exist apart from the mathematics tutor. He might excessively 
merge with the tutor. If so it will be the tutors' role to mirror his emerging mathematics 
competence back to him and to provide timely tolerable frustrations. Then the student can 
begin to discover his own prowess and learn to do mathematics on his own, with 
appropriate support. Alternatively, if he believes, even unconsciously, that he is incapable 
of imderstanding the mathematics he may try to memorize all procedures and will likely 
defensively blame external factors when this is unsuccessful. Such a student may avoid 
seeking help from teachers or tutors citing lack of interest or effort as the reason for his 
lack of success. 
Attachment Theory and Depression 

Not all mathematics depressions are rooted in underdevelopment of self By 
contrast, a student may be in the depression stage of a grief process. A student who is 
used to doing adequately but then experiences doing badly may be thus affected. Another 
student who has experienced a teacher's dislike, rejection, or humiliation after a history 
of positive teacher experiences may also feel depressed in a subsequent mathematics 
classroom unless he is helped to work through his depression. 

This line of thinking about relational origins of depression had led me to seeing 
that and how assessment results and counseling interventions for mathematics 
depressions originating primarily in the self dimension (empty depression) might differ 



91 

from those that origuiated primarily in the interpersonal attachment dimension (grief 
depression). 

Grandiosity 
Self Psychology and Grandiosity 

Although U.S. students perform much less well than students from other 
developed countries, a persistent fmdmg in international mathematics studies is that, on 
average, they think more highly of their mathematics ability than do the students who 
outperform them (National Center for Educational Statistics, 1995, 1999). U.S. males are 
more likely than females to think more highly of their prowess than their achievements 
would suggest to be appropriate (Sax, 1994; Signer & Beasley, 1997). Struggling 
students with a grandiose (all-powerful, all-knowing) view of their mathematics 
functioning are rarely seen in the learning support center because they cannot consciously 
face a need to get the help they need. 

Grandiosity may be linked with an underlying poor mathematics self-esteem 
because of early teachers' failure to provide the student with the tolerable reality that the 
student is not all-powerful or all-knowing, even in his teacher's eyes. This in turn leads to 
inadequate internal mathematics structure and values needed to curb his grandiosity via 
idealizing and incorporating his mathematics teacher image. He is likely to deal with a 
mathematics class or a specific mathematics problem by expressing his belief in his 
ability to do it while he fails to put in the effort needed to succeed. He seems to be 
unwilling to risk putting in the effort and risking that he may not be able to do 
mathematics. That risk is too great for his vubierable and underdeveloped mathematics 
self, so he may preserve his unrealistic sense of his ability by doing poorly or failing the 



92 

course and attributing this to his lack of effort. The challenges a mathematics counselor 
might face in trying to help such a student seem considerable. The greatest challenge is 
persuading him to get help and the counselor has to be very careful initially to accept the 
student's grandiose view of himself while fmdiag ways to diagnose and remedy his 
mathematics gaps and deficits. 

Mathematics Mental Health Conditions: A Summary and Caveats 
When I considered the ways a student might present himself to a mathematics 
counselor, it was clear that the conditions I discuss above are far from exhaustive. Each 
describes a dimension or continuum of cognition or affect common to every student. 
Where a particular student's results are located in one dimension or combination of 
dimensions will allow the tutor to determine the state of his mathematics mental health 
from sound through poor. Whatever his state, growth is always possible. I expect that not 
only students who consider themselves poor at mathematics could benefit from engaging 
in this process of relational mathematics counseling. It was also clear, though, that a 
relational counseling approach ensures that even if a student comes to counseling with a 
condition different from those discussed here, the tutor will be able to understand him 
well-enough to help him understand himself and improve his mathematics mental health. 
I became mcreasingly aware that engaging in this process with a student likely involves 
not only his change and growth but also mine. The role of the instruments I developed or 
adapted must be adjunct rather than definitive; the role of the tutor and the relational 
counseling approach should be preeminent in the growth and achievement of both the 
tutor and student. 



93 

IMPLEMENTING A BRIEF RELATIONAL COUNSELING APPROACH 
Extreme scores on a student's Feelings and Beliefs in-class survey assessments 
may alert him, me, and/or his instructor to the possibility of his benefiting by counseling; 
he may have entered the course expecting to need assistance and comes early to seek the 
regular help he believes he needs; or he might be prompted to come by a quiz or exam 
grade below his expectations. Ideally he would begin a course of mathematics counseling 
early in the semester, enough before the first big exam that at least half of the first session 
could be devoted to gaining some understanding of his mathematical background and 
experiences, and his current sense of himself as a mathematics learner. Student 
expectations about how mathematics counseling might be similar and different from 
traditional mathematics tutoring might need to be discussed; students are unlikely to be 
consciously aware of the possible relationships between their mathematics relational 
patterns (mcluding their sense of mathematics self) and their approach and achievement 
potential in the current class. Reahstically, students are likely to exert considerable 
pressure to focus on the mathematics content of the course from the beginning so the 
process of gathering background information and the process of orientating them into a 
mathematics counseling approach will need to be ongoing through the course of 
counseling. 

In that first session, I would ask the student to create the metaphor whose threads 
and themes we could explore over the course of counseling. My curiosity about how he 
came to where he is now would also form a thread running through sessions as we pull 
apart the mathematics challenges he is facing. The counselor must be alert for his own 
reactions, and for behaviors in the student that could be elucidated by the student's 



94 

Feeling and Beliefs responses, his metaphor, his mathematics background and 
experience, and his present mathematics performance. Class assessment results, the 
student's responses to them, and the mathematics patterns they reveal are likely to be 
focal in counseling. The JMK Affect Scales filled out at each session would provide 
regular opportunity to explore links among behaviors, beliefs, and exam results. 

I realize that the student in mathematics counseling is part of a complex system of 
important players. Each, including himself, is faced with multiple roles. In my study I 
focus on the student and the counselor, but others, especially the instructor, play active 
roles the student's and the counselor's mathematics relational worlds. 
Roles Played in Mathematics Counseling 

In this brief relational mathematical counseling approach, it is not only important 
for the counselor to understand and integrate a great deal of information about the 
student, but he also has to consider roles of all parties: the tutor/counselor, the student, 
and other significant players (e.g., the instructor) within the college context. The 
mathematics counselor or the mathematics counselor and the student together become 
aware of the student's mathematics dimensional relationships as a vehicle for both to 
know the student holistically and identify what and how he needs and wishes to change. 
Approaches, assessments, and therapeutic contributions from each of three dimensions of 
the relationship have been identified. 

At this point I need now to discuss new and necessary orientations and 
preparation of a mathematics relational counselor. By definition, the counselor 
undertakes to view the student wdth unconditional positive regard. He imequivocally 
believes in the student's existing mathematical intelligence and the potential for that to 



95 

grow. The counselor must also understand the counseling and mathematics learning 

processes to be a collaborative effort. The counselor brings expertise in mathematics, in 

mathematics pedagogy, and in relational counseling approaches and techniques;"" the 

student brings his own reality, his mathematics understandings and potential and his 

willingness to explore, consider, and apply insights that emerge in the counseling 

process. 

Relational Counseling Role: The Therapeutic Relationship 
in Mathematics Counseling 

The following roles emerged for me as ideal yet potentially attainable: 
The Mathematics Counselor as Listener and Witness to the Student 's History 

The counselor listens knowingly (mathematically and developmentally) with 

curiosity rather than with judgment. He elicits the student's experience of his own 

mathematics history. To test the efficacy of interviewing for understanding students' 

mathematics identity (self) development, I developed a semi-structured interview outline 

and piloted it with basic algebra students at a small liberal arts college in the Northeast 

(see Appendix A). The interviews I conducted with these students about their 

mathematics identity development corroborate Buxton's"'" (1991) fmdings and is the 

protocol I developed that I use here (Knowles, 1 998). 

Transference and Countertransference in the Mathematics 
Counseling Situation 

The counselor must be alert to how the student responds to him as a significant 

mathematics figure from the past (transference); the counselor also watches for ways he 

unconsciously responds to the student's transference or as a significant figure from his 

own past (countertransference). This awareness and mterpretation of cUent transference 



96 

and counselor countertransference in the counseling relationship are central to relational 

psychotherapy. Close observation of the counselor-client relationship yields crucial data 

for identifying relational patterns that are either conscious or unconscious, and that can be 

either beneficial or counterproductive to the student's sound mental health. In the 

mathematics counseling setting this requires the mathematics counselor to become 

conscious of how he experiences the way the student relates to him and seems to expect 

him to be as a teacher (transference). He must also become aware of his own reactions to 

and hopes for the student, understanding direct responses to the student and knowing 

reactions that are based on those from his own teaching or other relational experiences, 

triggered by the relationship with the student (countertransference). 

Insight, Central Conflict Identification, and Interpretation 
in Mathematics Counseling 

The counselor observes and hears patterns and unconscious contradictions among 

aspects of the student's relationships that may help to explain the student's puzzling 

mathematics-related behaviors and may yield clues to identifying his central relational 

conflict (insight). He then discusses and clarifies these with the student (interpretation) so 

the student may gain insight into his problems; 

Mathematics Counseling Role: The Tutoring Relationship 
in Mathematics Counseling 

The counselor models healthy mathematical behaviors and interprets them in 

relation to his own underlying healthy beliefs about the mathematics, himself, and the 

mathematics learning situation. He cannot presume that the student will make these 

connections between behaviors and beliefs without sometimes extensive mutual 

interpretation. 



97 

Mathematics Tutoring as Central to Mathematics Counseling 

In standard relational counseling, the focus is both the client and his relational 

problems; in mathematics relational counseling the focus is the student and his 

difficulties learning mathematics. The counselor must balance therapy's client- 

centeredness and counseling's problem-centeredness (Corsini & Wedding, 1995) by 

adopting the dual focus of relational brief therapy. 

Contributions of Conceptual Mathematics Tutoring and Mathematics Course 
Management Counseling 

The counselor is an experienced mathematics learning specialist who is aware of 
the toxic effects of an exclusively procedural approach to mathematics and the 
importance of strategic course management in a time limited college course setting. 
Although these issues may not be focal in the eyes of the student, the counselor must be 
alert to any need to incorporate them into successful cotmseling, Understanding the 
motivations of the instructor (and the department) is also key since ambivalence about 
what is valued as mathematics outcomes and how the instructor assesses these outcomes 
may result in confusion for students between getting good grades and reall\- working for 
comprehensive understanding'^' (Hiebert, 1999; Lee & Wheeler, 1987; Mokros. 2000). 
The learning counselor typically has little if any influence on the curriculum or the 
assessment so his role is to help the student adapt to the course in a way that is as healthy 
as possible for him. 

"Understanding is an ongoing activity not an achievement" (Kieran. 1994. p. 589) 
but its hnks with mathematical self-esteem places the onus on the mathematics counselor 
to discern compromises between achievement (of grades) and understanding: In addition 



98 

the student needs effective ways to adapt to the present mathematics classroom at the 

same time that he repairs his mathematics self-esteem and succeeds in the course. 

Constructivist Approaches: The Student as Author of His own Growth and 
Healing in Mathematics Counseling Situations 

The relational mathematics counselor believes the student has what he needs 
relationally, intellectually, and especially mathematically to make the changes he needs 
in order to achieve good-enough results. The approach to the counseling and to the 
mathematics is thus a developmental constructivist approach. This, however, does not 
preclude strategic direct teaching in the time limited setting. 

Roles in Relation to Other Key Players 
The Student and the Instructor 

The student's relationship with his instructor is likely to be revealing not only of 
his present mathematics learning approaches but also of his historical patterns of relating 
with mathematics relationships. As the counselor becomes aware of the student's 
perceptions of the relationship with the current teacher and as they are both able to 
directly observe the relationship, the counselor may use discussions of the congruence 
between the two to explore these patterns. How the student perceives himself in relation 
to his classmates and relates (or not) with them is also likely to be of interest although not 
as pivotal as his relationship with the instructor. 
The Counselor and the Instructor 

Effective tutoring involves not only supporting students in learning the content 
covered in the syllabus but also in helping them understand the instructor's teaching 
approach, assessment schemes, and priorities. This implies the tutor's knowing or being 
able to understand the instructor's approach. A relational approach implies in addition 



99 

that the tutor/counselor know or be able to gauge how the instructor's pedagogy, 
classroom management style, and relational patterns might impact the student. The 
counselor must discuss the instructor's approach with the student (and possibly the 
instructor), especially if it seems to be detrimental to the student. Ensuring that this 
happens this is likely to be extremely important to the efficacy of counseling. The 
counselor has to be conscious of his relationship with the class instructor and may have to 
use this awareness in mathematics counseling to help the student fmd ways to negotiate a 
constructive relationship with the instructor and class. 
Supervision of the Counselor by a Person Knowledgeable in Counseling 

Because a major source of insight for the counselor is the transference and the 
countertransference in the counseling situation, he should be under supervision. This 
means that at least once or twice during the semester he should present himself and his 
student as cases to a person knowledgeable in counseling in order to confirm or challenge 
his insights and approaches and to gain insight and inspiration in cases that he continues 
to fmd puzzling. 

CONCLUSION 

I have situated brief relational mathematics counseling in the college learning 
center context and pointed to the details of what it might look like. I have designed a 
summary chart that illustrates its components and how I see they relate to each other (see 
Table 3). What follows in this dissertation is a description of what happened in the pilot 
study as I applied the theory explored in chapter 2 in ways that I envisioned in this 
chapter. 



100 



In the next chapter I will describe the research methods I used to describe my 
pilot implementation of brief relational mathematics counseling with students in a 
statistics for psychology class at a small university college in the Northeast. 



101 



' Because this is an odd numbered chapter 1 use "he," "his," and "him" as the generic third person singular 
pronouns. 

" Through intelHgence testing, "aptitude" tests such as the SATs (Scholastic Aptitude Tests), or 
standardized achievements tests with percentile rankings interpreted as ability measures. 

"" Through school mathematics grades and "ability" grouping, and teacher/school and parental/societal 
expectation. 

'" As noted in chapter 2, people suffering from depression have been found by cognitive therapists to view 
the world, themselves, and the future through a negative cognitive schema (Beck, 1977; Beck, Rush, & 
Shaw, 1979). Martin Seligman (1975) has shovm that a person suffering from situational depression has 
almost identical symptoms to those suffering from situational learned helplessness. 

" How he processes and assimilates new learnings, accommodates his cognitive schema to these new 
learnings, stores them in long-term memory, and retrieves them for application and in appropriating further 
learning, constitutes the student's cognitive learning style (Davidson, 1983; Piaget, 1985; Schoenfeld, 
1992; Skemp, 1987). Skemp (1987) uses Piaget's term "assimilation" as the process whereby the learner 
assimilates the new learning into existing conceptual schema and at the same time "accommodates" the 
existing conceptual schema to meet the demands of the new situation, resulting in a struggle to arrive at an 
expanded schema and new greater understanding. 

Davidson (1983) found strong links to hemispheric preference and clearly defined analytic (her 
Mathematics Learning Style 1) and global (her Mathematics Learning Style II) learning styles in terms of 
students' mathematical behaviors and approaches. Although these learning styles have not been found to be 
directly related to mathematical achievement, Krutetskii (1976) found that students who had a strong 
learning style preference and a relative inability with their other mode, found it difficult, if not impossible, 
to begin a problem using the other mode's approach. Thus being forced to approach problems using 
another's preferred approach greatly disadvantages these students. Learning flexibility, however, can 
strengthen performance. 

" A strong leaning towards one learning style/processing channel (e.g., visual versus auditory versus 
kinesthetic) may present as a learning disability in an environment where another is favored but may be 
celebrated as ability in an environment where the preferred style/channel is favored. Thus there are 
individuals for whom having a learning disability may be more relative to the learning environment than 
intrinsic to her. 

™ In the high schools, teachers may know the mathematics content, but the pedagogy is often teacher- 
centered and procedural, as procedural mathematics routines are transmitted to the students. Many 
members of the mathematics education community have come to believe that exclusive exposure to a 
transmission model of pedagogy is generally antithetical to the development of students' mathematical 
power. Unfortunately it is what most U.S. students coming to college have experienced (Boaler, 1998; 
International Association for the Evaluation of Educational Achievement, 2001; Skemp, 1987). 

™' Analysis of attempts to measure mathematics anxiety reveals problems in the research in understanding 
what exactly is being measured (McLeod, 1992). Use and development of Mathematics Anxiety Rating 
Scale (MARS) (Richardson & Suinn, 1972) which has been normed and is perhaps the most used in the 
field, is illustrative of the problem. MARS doesn't distinguish among different types of anxiety, for 
example, cognitive worry versus affective emotionality that some theorists differentiate (cf Ho, et al., 
2000) or between state versus trait anxiety (cf Nolting, 1 990). MARS defines anxiety by a single affective 
response — fright — and asks students to distinguish among five levels of fright in relation to mathematics- 
related activities and situations. Factor analysis of MARS items yielded two relatively homogeneous 
factors (15 items each): mathematics testing anxiety and numerical anxiety (Rounds & Hendel, 1980). An 
additional factor, abstraction anxiety, important for college students but not addressed in MARS has been 
identified by Ferguson and a resulting three scale (the first two of which use MARS items identified by 
Rounds and Hendel) test — Phobus — developed (Ferguson, 1986). The last two scales of Phobus (numerical 



102 



and abstraction anxieties) difierentiate between two types of mathematics, each, however, in different 
settings: number/arithmetic (outside the classroom in every day settings) versus mathematics involving 
algebraic variables and other literal symbols (in classroom and college settings). The first scale inquires 
about mathematics testing-related situations before during and after the test. Items in Phobus can be further 
classified according to whether the activity is likely to be solitary or public or either. No items inquire 
about effects of degree of test preparation on anxiety levels nor do these scales ask about effects of anxiety 
on cognition during testing. A deficits model of testing anxiety proposes that a student who is poorly 
prepared and has poor test-taking skills will have high anxiety in testing situations (e.g., Tobias, 1985), and 
an interference model of testing anxiety proposes that in testing situations, anxiety interferes with students' 
recall and thinking. 

" Freud (1926) believed separation anxiety to be a natural response to separation and loss only in children. 
In adults he viewed it as pathological. 

" Only 1 1% of the 157 above-average college students seeking certification in elementary education, 
surveyed, reported only positive experiences in their own mathematics education. Of the others, when the 
onset of their anxiety was in the 3"^ or 4* grade (as for 16% of the sample), among behaviors of instructors 
cited, instructors were perceived to not respond to students' needs for clarification and tutoring or showed 
anger or disgust when students asked for help (p. 584). Many of the students whose negative experiences 
began in high school (26% of the sample) reported the same ignoring, rejecting, or ridiculing of students' 
needs, as did many of the 27% of students whose problems began freshman year of college (p.584). 

" This is a version of Kogelman and Warren's (1978) Myth 1 1 : Some have math minds and some don't; or 
Schoenfeld's (1992): Ordinary students cannot expect to understand mathematics; or the National Research 
Institute's first Myth: Success in mathematics depends more on innate ability than on hard work (National 
Research Council, 1991, p. 10). 

™In addition to access to a supervisory person knowledgeable in counseling psychologies, for a 
professional mathematics tutor to engage in the brief relational mathematics counseling described here, 
some preparation (i.e., coursework or at the very least, directed reading) in counseling psychologies, 
including CT/CBT and relational conflict therapy would seem to be a minimal requirement. 

'"' To understand (and ameliorate) adults' mathematics "panic," Buxton (1991) looked at individual in- 
depth interviews, group study of mathematics problems, and discussions of affect. His participants' stories 
invariably linked their mathematics panic and failure to achieve to parents, teachers, and their theories 
about themselves. What they believed mathematics to be and how they experienced mathematics teaching 
in their lives interacted significantly vnth their mathematical self-perceptions. 

"Understanding can be characterized by the kinds of relationships or connections that have been 
constructed between ideas, facts, procedures and so on... there are two cognitive processes that are key in 
students' efforts to understand mathematics — reflection and communication" (Hiebert et al., 1997, p. 15), 
both of which require the opportunity and time to do so. 



103 

CHAPTER IV 

METHODOLOGY TO STUDY BRIEF MATHEMATICS RELATIONAL 
COUNSELING MODEL 

LEARNING ASSISTANCE CENTER AND MATHEMATICS SUPPORT: 
Finding an Appropriate Research Setting to Pilot Brief Mathematics Relational 

Counseling 

My responsibility as the mathematics specialist for the Learning Assistance 
Center at Brookwood' State University is to offer support to students taking mathematics 
and mathematics-related courses. Along with the mathematics peer tutors whom I help to 
train, I offer mathematics tutoring in individual appointments, in open drop-in tutoring at 
the Learning Assistance Center," and in study groups for specific courses. These 
offerings are advertised to students via memos to instructors and by initial visits to the 
classrooms by the peer tutor or me. Some students who need support fmd their way to 
the Center in a timely fashion and often enough for the support to help; some come at the 
last minute (e.g., just before an exam when it is often difficult to resolve their problems); 
others do not come at all. 

We do not see all of the struggling students and generally cannot directly observe 
how the students we do see are handling their instructors' teaching and testing 
approaches. For my study of whether and how relational counseling insights could 
contribute to mathematics support, however, it was necessary to observe the classroom 
environments and student behaviors and interactions there. I decided that my research 
should focus on one mathematics course so that I could attend that class and offer 
individual mathematics counseling to its members. This approach was modeled on a 
tutoring practice already used in writing-intensive classes at Brookwood, where peer 
tutors are class-support tutors'" who attend the assigned class and offer learning 



104 

assistance both within and outside the classroom. Although this approach had not yet 
been used in mathematics classes, it was a familiar practice at Brookwood in other 
subjects, and the advantages for my research seemed obvious. 

To pilot my counseling approach I decided to focus on students in a class that was 
considered to be at risk for high negative mathematics emotionality, withdrawal, and 
failure. The PSYC/STAT 104'^ (Statistics in Psychology) class that I researched was a 
one-semester introductory statistics course that fulfills the university's quantitative 
reasoning core requirement. It is also a major requirement for nursing, psychology, and 
biological science students. This course is taught for fifteen weeks in the fall and spring 
semesters and for ten weeks in the summer" at Brookwood State University. Ann Porter"', 
the PSYC/STAT 104 instructor, a tenure-track faculty member, agreed to host this 
research in her classroom. 

The specifics of the course, the students, the instructor, and the mathematics 
counselor were particular to us. However, I was certain that my observations, diagnoses, 
and the application of brief relational mathematics counseling approaches to Brookwood 
students' challenges would provide insight into some broadly applicable ways that 
mathematics students can be supported and shed light on changes needed in traditional 
college mathematics support. I expected a fi-amework could emerge to help mathematics 
support professionals to understand and deal with students' mathematics problems in a 
way that would also promote their mathematics mental health while they are engaged in a 
semester course. The emergent framework is grounded in relational conflict brief 
counseling theories and cognitive constructivism. 



105 

The Course 

The introductory level statistics courses, PSYC/STAT 104 and BUS/STAT 130, 
are among the most failed and dropped first year college level classes at Brookwood."" 
Although there is no stated mathematics prerequisite for PSYC/STAT104, there has been 
ongoing pressure from academic counseling and academic support personnel on the 
Enrollment Management Committee to make successftil completion of high school 
algebra at least a strong recommendation. 

PSYC/STAT 104 is offered through the psychology department. Some students 
take it to fiilfill the quantitative reasoning requirement for a liberal arts degree. Nursing 
and psychology majors are required to take it. Nursing faculty see it as a gatekeeper 
course for the degree: If a registered nurse (RN) is not able to pass it with at least a C, it 
is thought that she"" might not be a suitable candidate for a bachelor's degree. 
Enrollment in the summer course is always lighter than in the fall/spring semester courses 
and the course only takes 10 weeks to cover 15 weeks of material. During the summer, 
students typically work full-time and take PSYC/STAT 104 and at most one other course. 
The class offered in the summer of 2000 was typical, with RNs, psychology majors, and 
others, all hopmg to do well enough to be able to proceed towards their larger goals. 
THE STUDY SITE AND PERSONNEL 
The University 

Brookwood State University (not its real name) is a small commuter university 
college with approximately 1,500 degree and continuing education students. The summer 
enrollment is approximately 550. It is located in the small New England city of 
Brookwood. The greater Brookwood area population is almost 200,000 and is 



106 

predominantly white with 5.6% non- white or mixed race residents concentrated in the 
city proper. It is ethnically quite diverse. Thirty-one and one half percent of the 
population has French or French Canadian ancestry and many maintain their ancestral 
language and culture. Three percent identify as Latino and 14.4% of the population 
speaks a language other than English, two-thirds of which are Indo-European languages. 
There are more than 50 different languages spoken in the local schools. Six point six 
percent of the population is foreign bom, half of these having entered the U.S. since 1990 
(U.S. Census, Census 2000). Among these are considerable numbers of reiugees from the 
Balkans, Africa, and the Middle East. The university's college credit and intensive 
college-preparation summer English Speakers of Other Languages (ESOL) courses 
attract between 30 and 50 high school students and adults per year, approximately 35% of 
whom go on to degree programs in the university. 

The average age of undergraduate students attending Brookwood is between 26 
and 27 years and the student population approximately reflects the racial and ethnic 
diversity of the greater Brookwood area (Brookwood University records, March 2003). 
The university offers two-year associate's degree, bachelor's degree, and some master's 
degree programs. Many students enroll in credit courses as non-matriculated, continuing 
education students. 

Until recently, many of the classes (including PSYC/STAT 104) were held 
downtown at the Riverside Center, while the Learning Assistance Center and the 
Computer Lab were located at the Greenville Campus on the edge of the city about five 
miles from Riverside. 



107 

The Researcher and the Course Instructor 
At the time of the study, I had been in the field of mathematics education for 
abnost thirty years, the previous 1 2 at college level. My professional focus had been on 
the teaching and learning of developmental and first-year college-level mathematics, 
although I had tutored students and trained tutors across the undergraduate mathematics 
spectrum. I had developed curriculum and placement testing and had taught mathematics 
courses at a community college and small four-year liberal arts colleges in New England. 
I had worked in academic support for these courses with general student populations and 
special populations that included learning, sensory, and physically challenged students. 
I worked at Brookwood State University as the mathematics learning specialist 
and assistant director of the Learning Assistance Center. Prior to the study, I had only 
briefly met Dr. Ann Porter (pseudonym), the course instructor, at all-college ftinctions 
since she worked almost exclusively at the Riverside campus. We communicated by 
memo and through peer tutors about study group and tutoring offerings for students in her 
courses. I had not met any of the students in the class except Pierre, whom I knew by 
name and face through the Learning Assistance Center's work with English speakers of 
other languages. 

I was aware of the negative reputation of the course among students who 
perceived themselves to be shaky in mathematics because I had tutored one third of the 
members of the PSYC/STAT 104 course'" at Greenville campus in the spring of 2000. 1 
was aware of another third who were struggling and I perceived the dread of students in 
the Learning Assistance Center who knew they would have to take it in the future. 



108 

Researcher as Mathematics Tutor and Counselor 
My role in this study would extend beyond that of researcher doing naturalistic 
observation to active intervention as a tutor and a counselor, so it was important for me to 
engage in continual self-reflection before and during the time of contact with the class 
and during the period of post-analysis of the data. In particular, 1 needed to reflect on 
myself as a mathematics tutor and also as an emerging mathematics counselor as I put 
relational and cognitive counseling theories into use. 

Who I am as a Mathematics Tutor 
I had always performed well as a student in the predominantly transmission'^ 
teaching, textbook-focused, and procedural"' mathematics classrooms of my elementary 
and high school education. At university, I became more conscious of the larger concepts 
underlying mathematics but it was not until I began tutoring students with learning 
disabilities that I became uncomfortable with the prevailing pedagogy and its implicit 
assumptions about students' learning processes; I realized how capable my tutees were 
but also saw how incomprehensible they found much of the mathematics presented to 
them in class. This began my struggle to understand their ways of thinking, to understand 
the mathematics more deeply myself, and to find ways to help them understand and 
achieve in a class that someone else is teaching, over whose curriculum or pedagogy I 
had no control. 

As a tutor I tend to help too much, by teaching and telling, more than to coach the 
student to find his own way to understand the material. I tend to suffer irom "agenda 
anxiety" on behalf of the students — knowing what they wUl be expected to cover but 



109 

worried that they might not recognize the urgency. I tend to try to push them too fast. I 
find it hard to let them make the mistakes they need in order to grow. 

I feel tension as a tutor of courses that other people teach, and this increases when 
the curriculum or the instructor's pedagogy seems to increase the students' difficulty in 
understanding the concepts and connecting related concepts. I feel even more tension 
when the student experiences the classroom as abusive or unsafe. At times I allow this 
tension to enter the tutoring session by siding with the student against the curriculum, the 
system, or the student's past preparation or teachers. I generally do not join students in 
criticizing the instructor but try to help them find ways to handle these conflicts in a way 
that is constructive to them. I sometimes find myself defending teachers whom students 
are attacking. 

Who I am as an Emerging Mathematics Counselor 

I am a white, university-educated, Australian, female, extroverted mathematics 
teacher and tutor brought up in the suburb of a large city in a middle class home by 
parents who were both tertiary educated professionals. In addition to working in 
mathematics education, I also worked with several groups of Australian Aborigines doing 
field linguistics and a trial literacy project. It was in the context of that work that I met 
and married a rural, working-class. New England American who graduated fi-om a 
technical high school program in the 1960s and works in the building trades. 

My faith is grounded in the principle that all humans are made in the image of 
God and are thus inherently creative and have the potential (indeed the obligation) to 
learn and grow and understand. It has been an important basis for my interest in and 
continually emergent acceptance of people whose backgrounds and characteristics are 



110 

different from mine. I am aware of my need to continue to work through my class-, 
ethno-, religion-, gender- and extrovert-centric orientations and grow in appreciation and 
acceptance of difference. As I explore and understand the challenges and opportunities 
that my tutees and my characteristics and backgrounds have placed on our development 
and our ability to understand and accept each other, I continue to find that some aspects 
of who I am provide potential bridges and others create potential barriers; I know I have 
blind spots that make understanding difficult. 

I am female, non- American and from what is often seen by Americans as an 
insignificant former British colony."" I live with a person from a low SES background 
who experienced low expectations and less education because of this background and still 
struggles with a sense of powerlessness. My efforts to accept and maximize my potential 
within these identities give me some empathy with students from disempowered 
groups — women, racial and ethnic minorities, and people of low SES — whose 
mathematics selves, internalized presences, and attachments have been negatively 
affected because of who they are."'" 

My struggles with arithmetic details (I cannot keep my checkbook straight), visual 
memory, visual-spatial reasoning, and directionality (I cannot tell left from right nor 
connect the implications of up versus down without verbalizing) enable me to empathize 
with students with learning disabilities or a strong mathematics learning style preference 
(and concomitant wealaiess in the other) who believe their learning challenges prevent 
their achievement in mathematics. I can also model struggle and success for these 
students. 



Ill 

My being white, middle-class, university-educated, and successfiil in mathematics 
may be initial barriers for students who feel disempowered because of who they are but I 
have found that self-disclosure of my struggles can help break through. My family 
background of trying to help a mother struggling with addictions makes me vulnerable to 
co-dependently take on a student's responsibility to make any changes she needs to, or to 
excuse his failure to take that responsibility himself I find the fme distinctions between 
support and indulgence difficult. On the other hand, I find it difficult to (and would rather 
not) work v\dth students who appear to be overestimating their abilities or knowledge or 
who seem to be rigidly adhering to approaches that are counterproductive. Hence, I 
recognize the particular importance in this study of attention to the student's transference 
and my countertransference in the mathematics counseling situation. 

The Instructor 
Dr. Ann Porter is a young, energetic, white woman (in her late twenties at the 
time of the study). She has a Ph.D. in experimental psychology and was actively engaged 
in research with a geriatric population at the time of the study. Ann also served as faculty 
advisor to the Student Government at the university. She had taught this course before. 
Ann began at the university two years before the study. She stated that she taught with a 
more "laid back" teaching style than she had experienced as a student (Interview 3, 
archived). As she described her professors' transmission teaching methods for her 
undergraduate introductory statistics and her later graduate statistics classes, she told me 
that she believed students should instead be able to grapple with the mathematical 
procedures during class with the opportunity to receive guidance rather than merely 
watch the procedures being done on the board as had been her own experience (Interview 



112 

3). She had been comfortable with algebra in high school, had minored in statistics in her 
doctoral program, and she was finding that teaching it to undergraduates was deepening 
her enjoyment of the field. 

Ann told the class she liked the assigned text, but disliked the required computer 
program MINITAB — a late 1 980s version — because it was "somewhat archaic" (course 
syllabus; Class 1, May 31, 2000). She told them of a more modem statistics software 
program she used to analyze her own data and promised to bring it in to show the class. 
She shared her own struggles with anxiety in a statistics class she had taken. She invited 
the class to call her by her first name if they preferred. All did. 

When I approached Ann before the course began she was hesitant about my 
doing research in her class because the class time needed to cover the material was 
reduced by several hours in the summer. "^^ She had committed herself to a very busy 
summer and she was also concerned that my research project would add to her 
workload. She was worried that my using a counseling approach with students for 
their "psychological" problems might have unforeseen repercussions on students' 
behaviors in the classroom and in relation to her, and make teaching the course more 
difficult. She did not see students' affective issues to be within her purview and did 
not want students to expect that of her. To allay her fears, I designed whole class 
research explanation and surveying to take minimal time and we agreed to schedule it 
just before breaks or after exams. Ann discovered her fears that my research would 
increase her workload were unfounded; indeed, the reverse was true. She found that 
in most cases students' negative affect and cognitive struggles actually became more 
contained because of the support I was offering. 



113 

My Roles in the Study 

I attended each class primarily as a researcher. In that capacity I took a small 
amount of class time to explain my project and administer pre- and post-surveys of 
mathematics affect to the class. Otherwise, I observed and recorded interactions in 
the classroom — especially instructor-student interactions during lectures and student- 
student and instructor-student interactions during problem- working sessions.'™ 
Increasingly I took the role of class-support tutor. In that capacity I led a weekly 
study group for the class and during class I assisted students sitting near me by 
working the problems in parallel with them, as Ann circled the room helping others. 
Ann occasionally consulted me on mathematical questions when she was uncertain. I 
also offered individual mathematics counseling to volunteers from the class and 
because students were meeting with me, they were generally less demanding of Aim's 
time outside of class. 

THE CLASS AND INDIVIDUAL PARTICIPANTS 
The PSYC/STAT 104 class of the summer of 2000 was typically small. There 
were 13 students (7 women and 6 men) at the first class meeting on May 31, 2000. 1 have 
given each a pseudonym to preserve anonymity. All were white and spoke English as 
their first language except Pierre, a French-speaking black African; at least one (but 
possibly three™) was first in the family to attend college, and most were long-time local 
residents. They ranged in age from 19 to the mid- forties, and about half traditional-age 
students. The class average age was around 28 years, somewhat higher than the 
Brookwood average. All but three were fiill-time students. Because it was a summer 
class, nearly half were from other colleges, a greater proportion than is usual in other 



114 



Table 4.1 

Profile Summary of Students taking PSYC/STAT 104. Summer 2000 (N = 13) 



Student 



Where 

Enrolled? 



Student-Related 
Data 



High School Math 
Courses 



College Math 
Courses 



Work in 
Summer 



PARTICIPANTS 

Eow^-R'' 
Autumn 

4 sessions 



SI/ 



W; Age ~20;Full- 
Time; MjrPsyc 
Mn'rBusiness to 
Mj':Business; Mn: 
Psyc 



Algebra I: A", Geom: Finite Math, 
A; AdvAlgebra II: C"; 2000: A" 
Prob&Stat(I/2): A 
Discrete Math( 1/2): A 



Retail~30hrs 



Ew-H? Brad BSl/ M"; Age~40s; Pt-time 

4 sessions Repeat PSYC/STAT 

104 



Algebra I, II, Geom: 
Bs?? 



PSYC/STAT, 
1998: F/AF 



Nursing-FT 



Ew-H Jamie 
5 sessions 



SU W; Age: 20; Full- 

Time; Mj: Psyc 
Repeat PSYQSTAT 
104 



Algebra I: B/B", PSYC/STAT, 

Geom: C"/D^ Algebra 1998: D" 

II: B7C*, Precalculus: Finite Math, 

C? 1999: W 



Retail~30hrs 





BSU 


W; Age: 22; Part-time; 


Algebra I; C? 


PSYC/STAT, 


Assistant 


Eow-H Karen 




Mj:Psyc, Mn:Educ 


Geometry (struggle); 


1998: F 


TeachCT - 


5 sessions 




Repeat PSVaSTAT 


Algebra II: C? 




Elem. Special 
Ed-FT 





SU 


W; Age:19;FuIl-Time; 


Algebra I; Geometry; 


Basic Math, 


Camp 


Ew-H Kelly 




Mj: LibArts; 


Algebra II 


2000: D/C? 


Counselor- 


3 sessions 




MjInt: Soc. Wk 






FT 


Eow-H Lee 


BSU 


W; Age:19;FulI-Time 


Geometry: B; Algebra 


Finite Math, 


Dental Office 


6 sessions 




Mj -Psyc 


II: A; Precalc/Calc: A 


1999: A 


Assist~30hrs 


Eow-H? Mitch 


BSU 


M; Age:23;Full-Time; 


Algebra I: F,A; 


PSYC/STAT, 


Retail: FT 


4 sessions 




Mj: EuropeHistory 
Repeat PSYC/STAT 
104 


Geometry: F,C 


1998: F 
Finite Math, 
1999: C 




Eow-R Mulder 


OU' 


M;Age:21;FuII- 


Algebra I; Geom; 


None 


Retail~30hrs 


5 sessions 




Time; Mj: Biology 


Algebra II:C 






Ew-H Pierre 


BSU 


M; Age~30s;Full- 


Algebra through 


Calculus I: D 


Residential 


8 sessions 




Time; Mj Biology; 
ESL/French 


Calculus 




Support: FT 


Eow-R Robin 


BSU 


W;Age~mid40s; Part- 


College Algebra A 


None 


Nursing: FT 


3 sessions 




Time; Mj: Nursing 








CLASS ONLY 




W;Age~40s; Full- 




Calculus I, 




(not individual 


BSU 


Time; Mj:Biology, 




1999: A 




participants) 




Mn:Educ 








Catherine 












EUen 


BSU? 


W;Age~30s; 










OC 


M;Age~20-30s; Full- 




Finite Math 




Floyd 




Time; Mj:Psyc 









Note. ''Participant signed up for mathematics counseling: Eow = Every other week; Ew = Every 
week. '' Participant's initial motivation for signing up was: -R = to help me with my research; -H 
= to get tutoring help; -H? = apparently to get help. '^Institution where student was enrolled: OC = 
Other College; BSU = Brookwood State University; SU = State U.; OU = Other U. '' Gender: W 
= woman; M = man; ^ Mj: Major, Mn: Minor; MjInt =intended Major; 'FT = full-time work. 



115 

semesters. Three of these were from the affiliated State University. All of the participants 
except Mitch had taken at least Algebra I, Geometry, and Algebra II in high school. Four 
students were repeating PSYC/STAT 1 04; three had taken the course within the past 2 
years and failed it (Karen, Brad, and Mitch) and one (Jamie) had earned too low a grade 
to be counted for her Psychology major. 

Ann briefly introduced me as a researcher at the first class meeting and in the 
second I gave each class member information about my research project (see Appendix 
D) and then administered class mathematics affect pretests (see Appendix C) to all who 
agreed to be involved in the whole class study. All 12 students present completed the 
pretests, thus constituting their consent to have me use them and classroom observations 
of them as data. How they could signify consent was explained in writing in the research 
explanation and I have archived pretests as consent agreements.'™" Ellen, who was not 
present, had dropped the course. At this class meeting all students were invited to 
volunteer to be individual research participants'"'" by signing up for one-hour 
mathematics counseling sessions with me. Nine students volunteered by fdling out and 
signing a volunteer agreement card. Four opted for counseling every week and five for 
every other week. One other (Lee) initially checked "no" for one-on-one counseling but 
e-mailed me the day before the first exam to ask to participate. Each of these signed an 
Informed Consent Form (see Appendix D) during the first counselmg session."'" Of the 
initial group often counseling participants, two failed to complete the course — one left 
before the second exam and another, citing family responsibilities, in the ninth week of 
the course. A summary of individuals' mathematics-related characteristics and history is 
presented in Table 4. 1 . 



116 



The benefit of participating in my study was the individual statistics tutoring and 

mathematics counseling, so monetary compensation for participants was not necessary. 

DATA COLLECTION 
The Research Schedule 



Table 4.2 



PSYC/STAT 104, Summer 2000 Class and Research Schedule 











Course Week 














1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


Post 


l"* Class 




Class 


Class 


Class 


Class 


Class 


Class 


Class 


Class 


Class 


"Comp 


Mondays 




2 


4 


6 


8 


10 


12 


14 


16 


18 


(extra) 


6:00 p.m. 




Pre- 
tests 


EXAM 

1 
June 12 

■PlB- 

SRA 






no 
class 

meeting 




EXAM 

3 
July 17 




MINITAB 

Post- 
Tests 


EXAM 


2""' Class 


Class 


Class 


Class 


Class 


Class 


Class 


Class 


Class 


Class 


Class 




Wednesdays 


1 


3 


5 


7 


9 


11 


13 


15 


17 


19 




6:00 p.m. 






MINITAB 

Cotnput 
Lab 




EXAM 

2 
June28 








EXAM 

4 
July 26 


EXAM 

5 
Aug. 2 




Study Group 




Study 


Study 


Study 


Study 


Study 


Study 


Study 


Study 


Study 




Wednesdays 
4:30 p.m. 




Gp 1 


Gp2 


Gp3 


Gp4 


Gp5 


Gp6 


Gp7 


Gp8 


Gp9 






4att. 


3att 


latt 


4/5att 


latt 


2att 


latt 


5att 


6+att 




Drop-In 






Ke,L 








B,L 


Ka, 
L, J 




Mu, 

MAT120 




Individual 






Ka6/12 


J6/20 


Ka6/26 


J7/3 


Ka7/I0 


Mu7/17 


Ka7/24 


P8/2 


J8/6 


Session 




TCe6/8 


A6/I2 


B6/20 


Mi6/26 


L7/5 


B7/10 


Ka7/17 


Mu7/25 


L8/2 


P8/7 








Mi6/14 

R6/14 

Ke6/16 


Mu6/21 


P6/27 


R7/5 


J7/U 


A7/I7 


R7/25 


P8/3 


L8/7 








L6/21 


A6/28 


Mu7/6 


Mi7/12 


L7/19 


B7/25 














Ke6/21 


Mu6/29 




R7/12 




P7/26 














P6/22 






P7/13 
B7/13 

P7/14 




J7/26 
Mi7/26 
A7/26 






My Outside 


Interview 

Ann 












Interview 

Ann 


Present 




Interview 

Ann 




activities 


Potter 












Porter 


cases 
to Dr. 




Porter 




in relation to 


May 












July 


P. 




Aug 




course 


31 












10 


July 20 

Study 

Gp 
before 
EXAM 
w/Ann 

4.6 pm. 
w/Jill 
5.*pm 




3 





Note. "After Exam #1, 1 administered the Statistical Reasoning Assessment (SRA); ''Students could take 
and optional comprehensive final after the course ended to replace a lower grade; '^A = Autumn; B = Brad; 
J = Jamie; Ka = Karen; Ke = Kelly; L = Lee; Mi = Mitch; Mu = Mulder; P = Pierre; R = Robin.; 
'The underlining in the table indicates the first individual mathematics counseling meeting for that 
participant. 



117 

Once participants had volunteered for individual mathematics counseling, we 
negotiated meeting times, and by the end of the fourth week of class I had met with each 
of the participants at least once for mathematics counseling (see Table 4.2 for complete 
schedule of the research). The number of counseling sessions ranged from three to eight, 
with an average of five per participant. 

I did not make my choice of individuals for the focal cases until after the course 
was completed so that during the sessions I would be equally focused on all 1 
participants. I audio-recorded counseling sessions and had ones I identified as key 
transcribed. My roles in sessions varied with the participant and the timing of the session 
(e.g., the proximity of an exam). 

Mathematics, Affect, and Relational Data Collection and Use 

Instruments for Assessment and Treatment 
Because I was piloting the brief relational counseling approach, I knew I must 
identify students' relational patterns and both affective and cognitive symptoms to be 
dealt with in the brief time available. I devised, adopted, and adapted a number of survey, 
emotional response, and mathematics cognition instruments, some of which I 
administered to the whole class and others to individuals in counselmg sessions. In 
chapter 3, 1 discuss my development and choices of individual instruments and indicate 
my proposed use in counseling (see Appendix B for the individual instruments). Also in 
chapter 3, 1 discuss my choices of class survey and mathematics instruments and indicate 
my proposed use of the instruments in counseling (see Appendix C for the class 
instruments). The individual case studies in chapter 6 reveal whether and how I actually 
used them in counseling. Chapter 8 includes an evaluation of the instruments' use in the 



118 

counseling process and there I make recommendations regarding their further adaptation 
and appropriate use. 

Mathematics Data Collection and Use 

I collected data about each student's mathematics skills using a statistics 
reasoning test (the Statistics Reasoning Assessment or SRA), administered to the class at 
both at the beginning and the end of the course, an arithmetic diagnostic (the Arithmetic 
for Statistics Assessment), an algebra diagnostic (the Algebra Test), all class PSYC/STAT 
1 04 tests,™ and participant-observation notes written during and immediately aiter 
classes and individual and group meetings (see Appendix C). With all but Autumn, the 
greater proportion of each session focused on the course's mathematics content. We used 
student class and assessment products to identify issues with strategic preparation and 
course management strategies. For example, exam analysis focused on accuracy of 
students' perceptions of what would be examined and how, their preparation, type of 
errors, and troubleshooting behaviors to enhance approaches to the next exam (see 
Appendix E, Table E4). 

Mathematics Affect Data Collection and Use 

Each student's conscious affect around mathematics learning was appraised using 
in-class pre- and post-feelings and beliefs surveys and discussion of his responses. The 
following is an example of how I used this survey data with a participant in a counseling 
session: When I pointed out Autumn's low score on the Learned Helpless/Mastery 
Oriented subscale of the Beliefs survey scale relative to the scale and to the class, she 
seemed a little surprised at first. Perhaps this was because she had offered to meet with 
me for my research and did not perceive herself to be in need of mathematics tutoring or 



119 



counseling. When I explained the concepts, however, she agreed that she had acted in a 
helpless way and at the same time revealed her disappointment with herself. It seemed 
that her performance learning motivation and her learned helplessness had conspired 
together to prompt her to a decision she later regretted. 

JK: . . . You answered in a way that seemed as if under certain mathematical 

situations, you would have a tendency to give up — 

Autumn: Oh, YEAH. 

JK: Or to not go ahead. 

Autumn: Yeah.t (laughs) 

JK: Okay. All right. That's— that's— 

Autunm: Definitely! 

Autumn: ... I remember now that when I was in 8th grade ... I was in the higher 
level math class ... But I was only getting 70s, and I wasn't happy with 
that so I wanted to go back, so I could get better grades. . . I went easier. 

JK: Easier class? 

Autumn: Because I'm a perfectionist, and that wasn't good enough. 

JK: . . . rather than going and seeing how you could get your grade higher? 

Autumn: Yeaht I just gave up and went down. 

Autumn: But I didn't challenge myself, so — (Session 2) 

Autumn had earlier expressed disappointment with her later mathematics achievement. 

Autumn: Um, mathematical achievement. I'm somewhat discouraged because I 
didn't really challenge myself enough in high school ... I kind of took 
the easy way out. 

JK: Ahh! So you feel that you could have achieved a higher level? 

Autumn: Yeah. I definitely could have. (Session 1) 

Autumn revealed in flirther discussion that she really was not "definite" that she 
could achieve at a higher level; she had "challenged herself and tried a harder class 
under difficult circumstances but had not gotten her required A. I surmised this was 
probably because she did not go for help, but she seemed to have decided it was because 
of an underlying inability to do harder mathematics — she took no further risks; al^er that 
she chose only classes she knew she could get an A in. Autumn had given in. In the 
conflict between safely preserving her high grades and achieving to what she hoped was 



120 

(but feared was not) her potential safety had won but Autumn was not happy. The learned 
helplessness discussion was fruitilil in two ways. First, it showed me that a student's own 
survey responses, while, in themselves, providing limited information, could form a 
stepping off point for both the participant and me to explore more deeply. Second, 
participants, at least in this case, will likely not reveal this type of information about 
themselves and their motives through direct questioning; use of their survey responses 
and my explanation of what these responses generally indicated about them seemed to be 
the prompt for such revelations. To support this conjecture, although I had asked Autumn 
about her mathematics course-taking experiences in Session 1 , she did not reveal her 
performance achievement-motivated (see chapter 3), course-switching behavior m 8* 
grade until Session 2 when I introduced her survey responses for discussion. 

Relational Data Collection and Use 

The principal means I used to collect data that linked affect and motives with 
relationality of which the participant was less consciously aware were through the 
metaphor and affect scales and through transference and countertransference. 
Metaphor and Affect Scales Data Collection and Use 

I gathered through individual metaphor surveys administered at the first 
counseling session and as part of the One-on-One Evaluation at the end of the course, an 
individual mathematics affect scales instrument administered at each counseling session, 
an individual mathematics learning history interview protocol, and classroom and 
individual meeting participant observation. Jamie was so unobtrusive in class that the 
instructor, looking back at the end of the course, wondered if she had started three or four 
classes later than the rest of the class (Interview 3). And she quietly slipped out of the 



121 



room whenever other participants were making their appointments. It was in discussing 
Jamie's metaphor that the role of her mathematics history and her personaHty in her 
present puzzling behaviors became clearer. 

JK: Yeah. And then what happens during the storm? How do you, like, 

handle the storm? 
Jamie: Um — stay inside, [both laugh] 
JK: So how does that relate to the math? 
Jamie: Um. Well, you have to prepare for tests. I don't know how staying inside 

does. (Session 1) 

Jamie went on in discussion to talk about an elementary teacher who had yelled and 

explained her reaction: 

Jamie: Yeah. You want to sit down and shut up so you don't bother her. 

JK: So maybe . . .you know, your presence in a classroom is very cormected to 

that? 
Jamie: Yeah.T (Session 3) 

Bringmg this history, her metaphor, and my observations of her current classroom 
behavior together enabled Jamie and me to realize that she was "staying inside" in this 
class almost as if she were still in her 5th grade class not able to do anything but survive 
the storm, but this behavior was jeopardizing her chances of success in the class. That 
conflict became our counseling focus. 
Transference and Countertransference Data Collection and Use 

I noticed transference and countertransference in individual participants' 
relationships with me as counselor and tutor. In some cases, we discussed it, providing 
data about participants' and my own subconscious mathematics-related relationship 
orientation. Here is an example of me slipping into a countertransference: 

JK: Maybe then your resistance is: you say, "This is conceptual. I don't have 

to do that." Maybe if you could say, "Ah this is not conceptual." 
Rename it: "This is just mathematical." 

Mulder: Pain in the butt! 



122 



JK: Am I a pain in the butt? [startled] 

Mulder: No, that section of the test 

JK: Well, you are doing a nice job of resisting, which is good . . . ( Session 5) 

I was almost certainly included in Mulder's "pain in the butt" classification. Here, 
as in previous sessions, I was giving advice, trying to fix his problem for him like a 
mother of a child rather than trusting him or allowing him to find his own way, and 
Mulder was actually resisting my countertransference with his rebellious teenaged-son- 
to-mother transference "Pain in the butt!" as much as he was resisting the cognitive 
challenges posed by the multiple-choice questions. In previous sessions I had scolded 
him and pushed him to overcome his resistance to mastering the conceptual multiple- 
choice part of the exams. In this session I continued: 

JK: Come on! Keep going! You've got a bunch of these to do. You are 

really resisting very well! . . . And what it does to me is like I'm thinking 
this guy is so smart he could do so well and the mother in me comes out 
and it's like "If I could only persuade him." 

Mulder: Yeah, I don't think you can do this one. [ignoring me] 

Mulder: Hey, I'm done, I'm done. 

JK: Oh, but look — there are these. 

Mulder: Oh, YEAAAH! Right on!! [very sarcastically] 

JK: There are not too many! 

Mulder: You make me really not want to come back here. (Session 5) 

When I recognized Mulder's "teenaged son" transference and admitted to my 
countertransferential indulgent but thwarted mothering approach I was able to recognize 
the inappropriateness and ineffectiveness of this cycle we were in and soon after, I 
removed myself from the cycle so Mulder could focus on his mathematical challenges 
instead of on the power struggle with me (see chapter 6 for flirther elaboration). This 
excerpt is an example of the transference and countertransference data collected and 
shows how I used my understanding of the transference and countertransference in the 
counseling process. 



123 

After each counseling session I examined and filed dated products from the 
session and completed a Mathematics Counseling Session reflection (see Appendix B). 1 
noted transferential, countertransferential, and relational dimension incidents. At the end 
of each day I audio-taped flirther reflections on the class, individual counseling sessions, 
study group, or other interactions that occurred that day. 

Efforts to Obtain Triangulation of Data 

Because much of the data I was gathering was subjectively experienced, and 
because understanding the interrelationships among data was essential for effectively 
helping participants' progress, I determined to work with a supervising counselor. There 
were several participants with whom I was struggling, and my own blind spots were 
almost certainly preventing me from seeing difficulties with others. After I had met 
several times with each participant, I met with a psychological counselor. Dr. P., 
presenting each participant and my experience of her for clinical supervision, for an 
expert perspective on subjectively experienced data and my responses to it, and for 
support and suggestions for ongoing counseling interventions. 

This meeting served the purpose of supporting, challenging, and focusing my 
emerging counseling efforts with participants. It also served as a key triangulation tool 
for the case study data, that is, it ensured that each participant's and my relational data 
was experienced by another knowledgeable person who actively participated in the 
relationship. The relational dyads between each participant became triangular — among 
each participant, me and Dr. P. Dr. P.'s later responses to my analyses of courses of 
counseling when the pilot study was completed further supported this triangulation 
purpose. 



124 

Triangulation was also provided by the instructor's perspective on the progress of 
the class and individual students. By the design of the study, in order to ensure that 
students' course outcomes not be compromised, the instructor was blind to survey and 
counseling data students gave me. I interviewed her before, during, and at the end of the 
course to learn her perspective on her teaching, on the students in her class, and on the 
effects of my presence in the class. Correlating her and my experiences of the classroom 
provided valuable insights into students' processes and changes in the classroom and 
assisted the progress of counseling. All mdividual meetings, the supervision session, and 
interviews were recorded on audiocassette. One class and the lecture portion of another 
were video-recorded. All material is archived. 

Data Collection Summary 

By the end of the summer course I had collected approximately 75 hours of 
audiotaped data from 48 counseling sessions and nine study group meetings, and an 
additional 25 hours from interviews, the supervision session, and my after-class 
reflections. I had 56 class exams (all the exams taken by each class member) 
approximately 20 completed pre- and post- feeling and belief surveys, 36 mathematics 
assessments and approximately 50 in-counseling affect/relational assessments from the 
ten participants. In addition, I had ahnost 100 pages of divided page course and 
observations notes, and approximately 40 class seating, lecture interaction, and problem- 
working session interaction charts. I also had copies of Ann's worksheets and the 
worksheets I devised for use in counseling. 



125 

ANALYSIS OF DATA 
Mathematics Educational Analysis 

Increasingly, mathematics education research recognizes the value of carefully 
conducted qualitative studies of teaching and learning processes and outcomes (McLeod, 
1997). In this study, a case study analysis based on the considerable amounts and levels 
of qualitative and quantitative""' data that I gathered and analyzed, best served to 
illustrate the model of brief relational mathematics counseling, developing a full picture 
that allows both researcher and reader to generate hypotheses that may be tested by 
ftirther cases and more experimental approaches. 

Despite the quantity of quantitative data I compiled, the conclusions I draw are as 
much those of a therapist as of a social scientist. Fundamentally, my research is an 
exploration of students' subjective experiences of mathematics, and of my subjective 
experience as a tutor and counselor helping her. In the words of Pierre Dominice, "The 
scientific model we have tried to respect in the educational sciences does not allow us to 
explore the vividness of subjectivity" (Dominice, 1990, p. 199). The scientific 
experimental method is not usually possible with human participants because all 
variables except the being investigated cannot be held constant. Therefore, in education, a 
quasi-experimental method is frequently used. Studies using this method attempt to hold 
constant as many variables as possible while causing the one or two variables in focus to 
change. In such studies, a multitude of variables, complex interrelationships among 
variables, the uniqueness of each participant, are all seen to be difficulties or variables to 
be reduced or at least evened out as much as possible to produce the uniformity necessary 
to show the effects of one variable on another. 



126 

In this study, however, not only do I choose not to ignore the complexities of 
interactions among variables and the uniqueness of each participant, but I embrace them. 
Complex human beings struggle with the influences of their conscious and subconscious 
existence on current mathematics practices and outcomes; simplicity would be a 
reduction of their educational reality. Case study analysis using both qualitative and 
quantitative analytic techniques is therefore the optimum choice. Nevertheless, fmdings 
from some elements of this pilot study may lead to the need for future quasi-experimental 
studies to establish their effectiveness. 

Dynamic Psychological Counseling Analysis 

Psychotherapy research offers this justification for the case study method: 

[T]he primary means of clinical inquiry, teaching, and learning has been and still 
remains the case study method grounded in the tradition of naturalistic 
observation. Statements about psychotherapy that are derived from group data 
typically have little direct relevance for clinical problems that are presented to the 
psychotherapist.™' (Jones, 1995, p.99) 

Advances in quantitative methodology in single-case research are leading to greater rigor 

and greater generalizability of the fmdmgs from such single cases (Jones, 1995). 

Additionally, psychoanalytic research tradition development in standardizing 

interpretation and treatment of clients' core relational challenges while taking care not to 

minimize the uniqueness and complexity of each person seem to me to be directly 

applicable to the mathematics counseling setting (Kemberg, 1995; Luborsky, 1976; 

Luborsky & Luborsky, 1995). 

Counselor-participant match is an important factor not only in counseling efficacy 

but also in psychoanalytic research analysis of counselor and participant insight, 

interaction, and change (Kantrowitz, 2002; Kemberg, 1995). Counselor-participant match 



127 

can be assessed in terms of particular conflicts that arise and more importantly, in terms 
of characterological similarities and differences that may hinder or support participant 
progress. In this study, the same mathematics counselor (I) met with each often 
participants involved in the same focal endeavor — the PS YC/STAT 1 04 course. These 
constants thus reduce to manageable proportions charting individuals' progress and 
comparing their issues and changes through counseling. In psychotherapy, supervision is 
considered crucial for helping counselors to identify blind-spots in their 
coimtertransference (Kantrowitz, 2002). Because of the number and variety of 
participants in this study, as discussed above, I turned to supervision by a person 
knowledgeable in counseling psychology to help me become aware of patterns of 
relationship with participants that were helpful for some but coimterproductive for others. 

The patterns that emerged helped me identify characteristics and mathematical 
relational patterns of students who ehcited similar or different countertransference 
reactions in me. For example, the motherly reaction that Mulder elicited in me was 
different from the one that Jamie elicited. I responded to Jamie with a nurturing, 
controlling mothering reaction after I had overcome her "mathematics teachers are 
dangerous; stay away from me" transference. On the other hand I responded to Mulder 
with an indulgent but thwarted mother countertransference. A key part of my method in 
relation to these focal participants was to analyze the countertransference they evoked in 
me. It became clear that a focus of study was the student-counselor dyad rather than the 
student or the counselor separately. 



128 

Integrated Cognitive and Relational Analysis 
Used in this Study 
Analysis of each participant's data and my relationship with him was ongoing and 

evolved through the summer. I mapped the mathematical and emotional paths the class, 
the individual students, the instructor and I walked, using data gathered principally to be 
analyzed and used with students during the study to inform the direction of their 
mathematics counseling. Data were also used in post-analysis of the study and in post- 
analysis of the effects on participants during mathematics counselmg. 

During the course, I studied the audiotapes, observation notes, and student 
products continually so as to develop strategic cognitive, affective, and relational 
interventions. 

Relational Episode Analysis 

A focal unit of study was the mathematics relational episode (cf Luborsky & 
Luborsky, 1995, and see Appendix E). Each episode was analyzed and triangulated with 
other data to determine what it revealed about the participant's central mathematics 
relational conflict. In acknowledging that it is her unresolved mathematics relational 
conflict that is preventing her desired achievement, it is important to see that this means 
that the student is struggling with a conflict of which he is only partly conscious. She is 
likely then to say and do things that appear contradictory, but it may be these very 
contradictions that reveal most about his central conflict (see Appendix E).To identify 
this central mathematics relational conflict, relational episodes were juxtaposed that 
revealed insights into each of the participant's three personal dimensions identified by 
Mitchell (1988): the mathematics self, internalized presences, and interpersonal 



129 

attachments. In Appendix E, I provide a discussion of analysis categories and the 
procedures and theory used to develop them. 

Conversation Analysis 
In order to communicate what transpired in a mathematics counseling sessions I 
needed to find ways of coding session transcripts to not only indicate transcription 
technicalities such as impossible or uncertain transcription, but also to indicate a sense of 
timmg, emphasis, and degree of agreement, and to allow for explanation of concurrent 
activity. I found some of the conversation conventions developed by Deborah Tannen 
(1984, p. xix) and those used by Anne Dyson (1989, Figure 1.1., p. 4) to be usefiil. I 
developed some of my own for functions they did not address, modified some where their 
Table 4.3. 
Conventions used in Presentation of Transcripts 



t marks enthusiastic agreement with other speaker 

■I- marks hesitant or minimal agreement with other speaker 

= marks somewhat agreement with other speaker 

(+) marks positive affect in tone of speaker 

(-) marks negative affect in tone of speaker 

I marks a glottal stop or abrupt cutting off of sound. 

NO that is, capitalized word or phrase, indicates increased volume. 

{ } includes parallel or immediately contiguous speech of the other person of the 

counselor-student dyad. If it is a person other than the counselor-student dyad 

speaking, that person will be named. 
* * indicates intentional waiting or pause time. 

indicates omitted material. 
/ / with no text included indicates that transcription was not possible 
/ / with text included indicates uncertam transcription. 
( ) includes notes referring to contextual and nonverbal information, for example 

(laughs), (surprised), or (unconvinced). 
[ ] includes explanatory information inserted into the quotation later by me. 
[I use conventional punctuation marks (periods, question marks, exclamation points) to 
indicate ends of utterances or sentences, usually [marked by conventionally agreed 
intonation changes and] slight pauses on the audiotape. Commas [indicate] pauses within 
sentence units. Dashes (— ) indicate interrupted utterances (Dyson, 1984, Figure 1.1., p.4). 



130 



distinctions were too fine for my purposes, and changed some for ease of word 

processing (see Table 4.3.)- 

Mathematics Behavior and Product Analysis 

I developed different coding categories for student verbalizations and behaviors 
during the major different in-class experiences. From analysis of the class lecture session 
data I developed the following coding categories for student questions, answers, and 
comments: (a) timing,'™'" (b) accuracy/relevance, (c) topic,'"™ (d) level of certainty 
(affective and cognitive), (e) frequency, and (f) development. From analysis of student 
behaviors in problem-working sessions, I developed the following coding categories: (a) 
topic/task, (b) seating, (c) tools, '"^ (d) interaction with instructor, and (e) interaction with 
researcher. From class exam data I developed coding categories for individuals and for 
the class: (a) pre-exam input (class treatment, student reaction and counseling 
preparation), (b) student's out of class preparation, (c) errors,""™ (d) trouble- shooting 
efforts, (e) instructor grading, and (f) post-exam counseling (see Appendix E for chart 
organizers of these coding categorizer schemes). What the analyses of class lectures, 
class problem-working, and class exams revealed about the student's personal 
mathematics relational patterns and central conflict I considered as the course 

Counseling Use of Analysis. 

As the study continued I devised ways to integrate data of different types so that I 
could use them to counsel participants and clarify their challenges. They were also used 
as interventions (e.g.. Survey Profile Summary Sheet, see Appendix B and chapter 6). 
With each participant I used insights and suggestions from the supervision discussion of 



131 

their data into following counseling sessions. The integration of data m supervision 
discussion increased my efficacy as a counselor. 

Post-analysis of all data, including participants' final evaluations and exams 
focused on relational episodes and their cognitive and affective links to relational 
patterns. The timing and fit of the ongoing analysis and the researcher's understanding of 
each participant's central relational conflict and related counseling interventions were 
determined. It was then that the three focal cases for deeper post-analysis and 
presentation were chosen in order to illustrate the brief relational counseling approach. 
BRINGING IT ALL TOGETHER INTO A CASE STUDY ANALYSIS. 
The PSYCH/STAT 104 Class as the Individual Case Context 

In chapter 5 I narrate the story of the class as a whole. That narration provides the 
basis for analysis of individuals' interpersonal relational patterns in the classroom 
context. Since the focus of this study is on the individual counseling and the student- 
counselor dyad, the particular value of examining the classroom context lies in the 
context it gives for the focal student case studies I present in chapter 6. In addition, when 
I conducted a comparative analysis of all participants' mathematics cognitive preparation 
and relationality, I expected a student classification to emerge not unlike Tobias' tier 
scheme. Tobias (1990; personal communication, March 16, 2001, May 20, 2003) 
formulated a tier analysis of science and mathematics undergraduates as they appear to 
academic support personnel. Given that Tobias's tier classification is accepted in the field 
of developmental mathematics education, it is, in a sense, the null hypothesizes 
classification scheme. As such I decided to use it for comparison purposes in describing 
the classification scheme that emerged from this study. In addition, and perhaps more 



132 

importantly, I considered that Tobias's tiers describe students she sees to be increasingly 
more vulnerable and in increasing need of academic support in order to succeed. It 
seemed advisable for me to take this into consideration in choosing my focal cases: When 
I choose from the ten participants, I chose students from vulnerable tiers. I had also to 
consider however that my study might identify other criteria that should influence my 
choice of focal cases. Tobias describes students in her tiers are as follows: 

The First Tier 

Students of the first tier are those who enter college well-prepared and confident, 
that is, with mathematical power (NCTM, 1989, 2000). They have developed conceptual 
understanding, are procedurally competent and are ready for new mathematical learning. 
Academic resource centers or mathematics centers frequently recruit mathematics peer 
tutors from this group. 

The Second Tier 

It is mostly the students in the tiers below who come to the attention of academic 
support personnel. Tobias identifies students in the second tier as capable students who 
have become convinced they "can't do mathematics." She observes that many of these 
students have learning styles different from the learning styles favored in the traditional 
mathematics classroom. They may be more verbal; they more often favor right-brain and 
visual thinking; and they are usually divergent thinkers and global (in contrast with 
analytical, c£ Witkin, Goodenough, & Karp, 1967; Davidson, 1983). It is not so much the 
mathematics subject matter but the pedagogy that has been the stumbling block for them. 
Depending on when and how these students experienced, "I can't do mathematics," they 
are more or less mathematically prepared. Almost invariably they believe they do not 



133 

have mathematical minds. Because most of these students are college bound, however, 
they may have struggled through three or even four years of traditional high school 
mathematics, often through precalculus. 

The Utilitarians ' Tier 

Students in the next tier, whom Tobias has designated utilitarians, have in her 
words "learned to play a mathematics game." According to her, they are procedural 
learners who are competent but not interested in understanding the mathematical 
concepts. They may have succeeded in traditional mathematics classes that emphasized 
procedural competence but may be unprepared for and resistant to the greatly accelerated 
pace and greater conceptual demands of some college mathematics courses. They may 
become angry if they fail or do poorly and they may be resistant to suggestions involving 
changing their ways of approaching mathematics. 

The Underprepared Tier 

In high school, many of these students were either not expected to attend, or did 
not intend to attend college, or if they did they did not expect to have to do mathematics 
in college, so they did little or no algebra. Others attempted some algebra in high school 
but were never engaged or did it a number of years ago. Still others "succeeded" in 
poorly taught or lower track classes. Whatever the reason, the underprepared have 
serious gaps in their knowledge base and often a poor mathematical self-concept. 

The Unlikelies ' Tier 
These are students Tobias designates as those "we can never reach." They 
include students who are hostile and "won't give us trust" (Tobias, S., personal 
communication, March 16, 2001). But with the "unlikelies" Tobias hesitates to cite lack 



134 

of mathematical ability as a cause of their difficulties and poor prognosis. Academic 
support personnel typically err on the side of faith in the ability of each student to 
transcend her difficulties, given the right combination of circumstances, change of heart, 
and support. However, most academic support personnel can point to students who would 
not or could not budge. In my experience, students least likely to succeed were those who 
are unable to confront their own difficulties honestly. 

Choosing the Focal Participants 

I chose three students, Karen, Jamie, and Mulder, for deeper case study analysis, 
using a number of criteria. My most important consideration was how their mathematics 
counseling illustrated different dynamics between the student and me involved in fmding 
a central relational conflict and how we used this insight to improve the student's 
mathematics mental health and success in the course. I also considered Tobias's tier 
analysis, however. With respect to Tobias's tier analysis, Jamie would probably be 
classified as second tier and Karen and Mulder had characteristics of the underprepared 
and unlikeUes, and, even in some senses, utilitarian tiers. Their stories are presented and 
analyzed in chapter 6. 

The focal participants were in many ways typical of students in need of support in 
their college mathematics course. Jamie and Mulder were traditional college aged, full- 
time students and had at least one parent who had a bachelor's degree; Karen was a little 
older, a part-time student, and the first in her family to pursue a bachelor's degree. Karen 
had previously failed the class and said she had always been poor at mathematics; Jamie 
had previously earned a D^ in the class and reported an uneven mathematics history, 
doing well or badly at different times. Mulder had not previously taken a college 



135 

mathematics course, and reported a history of not trying in high school mathematics 
classes and just getting by with Cs. 

These students reported family theories about their mathematics ability — Karen 
reported that hers was a reading and writing type family, Jamie said her mother's theory 
was that the women in her family were not good at mathematics, and Mulder speculated 
that he was probably capable of doing mathematics because his uncle and father were 
"smart." Not only were the focal participants similar and different in their histories, 
famihes and attributions, they also appeared immediately typically needy but for different 
reasons and in different ways from the perspective of the learning support center. In 
chapter 6 when I present the counselor-student dyad cases with Karen, Mulder, and Jamie 
I will discuss further these similarities and differences and their significance to my case 
selection. 

I present these students in the context of the class in the next chapter and zoom in 
on their courses of counseling in chapter 6 in order to illustrate the development and 
appUcation of brief relational counseling to identifying and treating central mathematics 
relational conflicts. 



136 



' I have given all institutions and locations mentioned in this study fictional names to preserve 
confidentiality. 

" The Learning Assistance Center has copies of all the mathematics course texts, student study guides, and 
student and instructor solution manuals. Instructors are requested to file their syllabi and class handouts 
with the Learning Assistance Center so that the peer tutors and I can keep pace with the courses as they 
progress through the semester. 

"' This class-support tutor has been variously labeled class-link tutor and class tutor. Typically this person 
would be a peer tutor (usually an undergraduate who has successfiilly completed the course), but it is not 
unheard of for a professional tutor to fulfill this role (M. Pobywajlo, personal communication, January 24, 
2000; Petress, 1999). 

" All course numbers have been changed to ensure confidentiality of the institution in which the research 
was conducted. The first digit used here is designed to indicate the level. For example, the number 104, 
with 1 as the first digit is a first year college level course. The course is described in the course catalog as 
follows: 

PSYC/STAT 104 (freshman level) 

Design, statistical analysis, and decision making in psychological research. Substantive problems 

as illustrations of typical applications and underlying logic. No credit for students who have 

completed BUS/STAT 130 or BIO/ST AT 105 (fulfills quantitative reasoning general education 

{core} requirement). Special fee. 4 cr. (From the on-line Brookwood State University Course 

Catalog) 

" The class was scheduled for Monday and Wednesday evenings 6:00 p.m. to 8:20 p.m. on the second floor 
of the Riverside campus building and ran from Wednesday, May 31 through Wednesday, August 2, 2000. 

" The names of all persons in this study have been changed to preserve their anonymity. 

™ The average attrition rate (drop, withdraw, fail) from 1995 through summer 2000 for PSYC/STAT 104 
was 26.6% over all. This breaks down to an average 3 1 .4% attrition rate for Fall/Spring semester courses 
and a much lower 14.75% attrition rate for the summer courses (archived grade reports, Brookwood State 
University). 

"" In keeping with my former practice, as this is an even numbered chapter I use "she," "her," and "hers" as 
generic third person singular pronouns. 

'" This course was taught by an adjunct psychology professor. 

" The teacher tells about and the students are expected to passively absorb the new mataial. See also 
chapter 2. 

" See chapter 2, endnotes xvi and xvii. 

"" Although Australians are considered racially and ethnically similar and are generally well-liked by 
Americans, there is an assumption that America and things American are bigger and certainly better than 
things Australian, and that Australia and therefore Australians are cute but inconsequential in anything that 
matters and are expected to agree and admire. I thus struggle with belonging in the U.S., with maintaining 
an Australian identity, and with feeling "less than" because of who I am. On the other hand, because 1 am 
not a vA\\te American, I am not implicated in the oppression of disempowered groups here (though 1 am in 
my country of origin). Now after 23 years here, I am more an Australian American than an Australian but 
continue to have a coimection with people who for whatever reason do not feel that they belong 
comfortably because of who they are. 



137 



'"" I realize that merely belonging to a disempowered group or being married to someone from such a group 
does not necessarily mean that I understand the challenges others from the same group face, nor how to 
encourage them to achieve their potential nevertheless. Indeed, for example, if one is at a low level of 
identity development, one is likely to buy in to the majority's negative assessment or low expectations of 
one's group and/or be trying to distance oneself from one's group and be trying to be like the majority 
(Ivey, Ivey, & Simek-Morgan, 1993; McNamara & Rickard, 1989). 

Americans have particular frouble with SES — few admit to having a low SES, that is, to belonging to 
the working class — and appreciation of values and cultures of the working class are rarely espoused 
(Frankenstein, 1990). The deficits are well-known: students from low SES backgrounds with parents who 
have not gone to college are less likely to go to college themselves or to succeed in college if they do. The 
Federal TRIO grant program provides extra support for such students in post-secondary education. My 
husband's experience of discrimination because of his SES background continues and we struggle with 
appreciating each other's different class sfrengths and weaknesses. Again identity developmental level (in 
this case class identity) is an issue, as is also an understanding of what might be involved (for my daughters 
and for my students) in learning about and negotiating the culture of power — the predominant culture in 
society and in academia (Delpit, 1988). 

"^ The summer 10-week session allowed for 4 hours and 40 minutes per week for 9 weeks and one class of 
2 hours and 20 minutes in the first week. Ann did not hold class on the Monday of the week of July 4. The 
total class time available was then 42 hours. In confrast, during a regular semester she would have between 
45 and 48 hours of class time to cover the same material. 

From Class 2 on, I used a music-scale like form and class layout form to record professor-student 
interaction for some portions of the lecture or lecture-guided problem portions of the class. During the class 
I noted the time at regular intervals during the class. I used these forms in subsequent classes and I 
developed an informal 2x2 charting procedure for diagramming interactions among students with each 
other and with Arm during the problem-working portions of class. After each class I tape-recorded my 
reflections on the class, professor, students, and on myself and my plans for the next class (I have archived 
these notes and recordings). See Appendix C for copies of the forms. 

™ Parents' college experience came up incidentally in counseling with some participants, but because I had 
incomplete data, I surveyed participants in November 2000 by e-mail. Of the six who responded, Karen, 
replied that her parents had not attended college. Lee's mother had an associate's degree from a technical 
school but neither parent had attended a four-year college. I believe that of the others who did not respond, 
Robin's parents (and possibly Kelly's) had not attended college. 

""^ I sought approval for conducting the research from both the Lesley University Committee on the 
Use of Human Subjects in Research, and from the Olfice of Sponsored Research's Institutional 
Review Board for the Protection of Human Research Subjects for the state university system to which 
Brookwood State University belongs, and was granted that approval. I have archived the official 
approval documents I received. 

^"' In this and following chapters I use the term "participant" to refer to students in the PSTC/STAT 104 
class who participated in individual mathematics counseling with me. 

"^ I have archived all original completed forms. 

^ Aim followed department policy in not returning exams to students. Instead she briefly went over exams 
in class with students and had them returned to her. However she agreed to allow me custody of exams to 
use with participants in counseling sessions and gave me all students' exams at the completion of the 
course. I have archived these materials. 

™ I determined that the primary use of new and adapted instruments would be descriptive; early use for 
individual affective, cognitive, and relational pattern recognition could be invaluable in helping the student 



138 



and me become aware and prioritize interventions. Already normed instruments might be usefial to develop 
realistic goals in the context of a course. Post testing using the instruments should give students indications 
of change in the factors surveyed, but the most concrete indicators of effectiveness of the mathematics 
counseling, for the students at least, would be improvements in exams or quizzes. Causal factors for change 
may be difficult to determine in such a study so hypothesized relationships among factors will need fijrther 
study. 

'""'In the past, research in psychotherapy into outcomes that involved pretreatment and posttreatment 
experimental designs resulted in findings that do nor account for the real complexity and non-linear 
experience personal processes. Research into process that involved time-sampling strategies and averaging 
of readily quantified units such as grammatical categories of speech produced findings that seemed 
disconnected fi"om the actual clinical experience and the theory behind the treatment. In any attempt at 
quantitative research, the problem of quantifying the "relationship between therapists and patients" arises 
but the fact is that this relationship "regularly appears in reviews as an important moderator of treatment 
effects" (Russell & Orlinsky, 1996) (p. 713). More recently, "researchers have turned to systematically 
conducted naturalistic studies to assess treatment effectiveness and clinical significance" (p.710). There is 
an important trend for researchers to "sift through the complexities of interactional and relational meaning" 
(p.71 1) and outcomes are being seen more as parts of a process rather than different phenomena. 

'°^ Timing is judged in terms of the extent to which the student's verbalization is linked in a timely 
manner with the instructor's utterance. For example, on a number of occasions Robin answered Ann's 
question with the correct answer to a previous question; her timing was off 

""^ Subcategories of topic developed were: (a) current content (mathematics; application; personal), (b) 
course strategy, and (c) grading. 

"" Subcategories of tools developed were: (a) text, (b) items provided by instructor, and (c) student 
provided aids such as calculator, notes, . . . 

■"^ Subcategories of errors developed were: (a) defining the problem: concepts, (b) planning the solution: 
procedures, (c) carrying out the solution: algebra, (d) carrying out the solution: arithmetic, (e) conclusion: 
Checking and reporting 



139 

CHAPTER V 
AN ACCOUNT OF SUMMER 2000 PSYC/STAT 104 CLASS 

In this chapter I will briefly reintroduce the students, introduce the physical 
setting of the classroom, and then discuss features of the class and teacher that were 
salient to the mathematics mental health of the students. Those include the curriculum 
and the text, the instructor's pedagogy, her view of statistics and mathematics, the 
emotional and mathematical climate established in the class, and how the students 
interacted with the mstructor and with each other. 1 will show how these features played 
out in the first few classes of the term and several typical or importantly different classes. 
From that picture, I will discuss each participant's experience of the class in relation to 
mathematics counseling interventions, highlighting the interactions among students' 
relational patterns and the classroom dynamics. 

Students 

The class consisted of 8 traditional aged students (18 through 25 years of age), all 
but one fiiU-time. The remaining 4 (5 if I include Ellen) non-traditional students who 
ranged in age from early thirties through mid- forties were part-time bachelor's degree 
students. Seven (or possibly eight) of the students were enrolled at Brookwood 
University; three were enrolled at State University; and the other two were enrolled at 
private colleges. All for whom I had data (I do not have that data on Catherine, Ellen, 
Floyd or Mitch) were working during the summer, five at vocational positions they 
maintained all year round, and four at temporary summer positions (see also chapter 4, 
particularly Table 4.1). 



140 

Students' had differing degrees of familiarity with the college mathematics 
courses. All had completed at least a year in college. Mulder and Robin were the only 
students in the class who had not taken a mathematics course in college. Of those who 
had taken college mathematics courses, only Autumn, Catherine, and Lee had been 
successfiil; the rest had either failed or earned Ds. 

Three students were repeating PS YC/STAT 1 04 because they had previously 
failed it in the summer of 1998. 1 found out after the study was completed that Jamie was 
repeating it because of a D^ on her first attempt in freshman year, not an acceptable grade 
for a course in her psychology major. Eight of the students were required to take 
PS YC/STAT 104 for their degree programs: Robin and Brad for nursing; Pierre and 
Catherine for biology; and Floyd, Ellen, Jamie, and Karen for psychology. For the other 
six, the motivation for taking the course was less clear. Two began the class with a 
psychology major in mind, thus requiring PS YC/STAT 104, but one changed her mind 
during the summer. The other began to waver on a psychology major, making it unclear 
whether PS YC/STAT 1 04 would be necessary for her. Another was taking it for elective 
credit to transfer. Mitch was taking the course to redress the messy situation of having to 
repair his GPA because he had failed it before, even though he said he believed that he 
should not have taken it in the first place. Kelly only needed to pass any college level 
mathematics course, something she had thus far failed to do. 

Only two students knew each other before the class began: Lee and Mitch. Mitch 
was the only student in the class who knew Ann the instructor, outside of the classroom 
setting; he was a member of Student Government for which Ann was faculty advisor. 



141 

Physical Settings 

The class usually met in a room on the second floor of the renovated former mill 
building that was the Riverside campus. The only classes not conducted in this room 
were Class 5, the MINITAB computer orientation class run by Aim and Pat, the computer 
lab assistant, which was held in the computer lab at the Greenville campus. Classes 4 and 
9 when Exams #1 and #2 were given were held in a classroom across the hall which had 
individual seats and attached desk-tops. Class 10 was not held as a class so that 
MINITAB project partners could meet during that week. 

Otherwise all classes were held in the same room. The space was almost entirely 
filled with six 2.5 by 5 foot tables arranged to make one 5 foot by 15 foot table, with 14 
or 15 chairs arranged around it (see Figure 5.1). 

I found that, although there was considerable variation in students' choice of 
seating, there were patterns that seemed to be connected to relational alhances, to 
technical constraints (e.g., Pierre's audio-taping), to the timing of a student's arrival, and 



I I chair 

2.5'x 5' table 



I><or left 




akboai 


d 














- 


right 





back 



Figure 5.1. Room and fiimiture configuration for PSYC/STAT104 class, second floor. 
Riverside Center, Brookwood State University, summer 2000. 



142 

to Other less obvious relational factors. Ann had previously taught only in classrooms 
with individual seats and attached desks facing the front and the chalkboard. She reported 
that her students consistently sat in the same seats. In this setting she was surprised by 
what she perceived to be almost random seating choices by students. 

My own choice of seating was largely driven by my desire to observe the class 
and individual members most strategically. I know that my seating choices affected 
students and their experience of the class and also undoubtedly affected what I saw of the 
class (particularly during problem-working sessions). In Appendix F I detail and discuss 
seating choices — both the students' and mine. 

PSYC/STAT 104: COURSE ELEMENTS AND EVENTS 
Class Presentation Organization 

Classes began at 6:00 p.m. The first part of a typical class consisted of Ann's 
presenting theory or as she said, "the concepts," with the overhead projector and the 
chalkboard. Aim always stood at the front of the room during the lecture portions of the 
class, moving from her notes on the table to the board or the overhead projector and back. 
The lecture took as long as the whole class period (i.e., from 2 to 2 hours and 20 minutes) 
if there was lecture guided problem- working interspersed but more typically went until 
break at around 7:00 or 7:15 p.m. (i.e., V* hour to 1 'A hours). Following the lecture. Aim 
usually handed out worksheet/s requiring the application of the theory just presented. 
She moved around the classroom checking over students' shoulders to see if they were on 
track. If a student seemed to be struggling, Ann would sit with him. She usually carried 
the worked solution so she could tell or show the students where and how their solutions 



143 

differed. When she had these worked solutions she would give me a copy so that I could 
help the students in the same way." 

Ann used arithmetical accuracy as a quick indicator of whether a student was 
proceeding correctly. When I had my graphing calculator with me I would use it to enter 
and analyze the data. Ann was unpressed with this as a quick way to fmd the 
arithmetically correct answers when she hadn't previously worked them out. 

When more than one topic was being covered during a class period, Ann typically 
lectured on one of the topics and had students do a worksheet that was sometimes lecture 
guided and sometimes done with her roving help. She then proceeded to lecture on the 
next topic, go to another worksheet and so on (Class 2, for example). During the lecture 
portion of the class Ann did not usually work problems on the board. Instead, she had the 
students use her worksheets, their texts and her over-the-shoulder help to work them out, 
sight unseen, during the problem-working portion of the class. In an interview, she told 
me that she this was a preferred method because it forced the students to find out how to 
do each problem themselves (Interview 3). 

The Curriculum and Textbook 

The text Ann used was Understanding Statistics in the Behavioral Sciences (5* 
edition), written by Robert R. Pagano (1998). It is an introductory non-calculus based 
statistics text using a typical sequence. The book treats descriptive statistics in the first 6 
chapters, followed by inferential statistics in the subsequent 12 chapters. Probability, 
random sampling and hypothesis testing concepts introduce the inferential section, 
followed by a "cookbook" of parametric and non-parametric tests. Ann's curriculum 
covered all but chapter 17 of the text, although some chapters were only partially 



144 

covered. Ann lectured from notes that kept quite closely to the text; at times she dictated 
directly from it. 

Students were expected to read ahead in preparation for the lecture and to practice 
procedures and solutions after the class.'" First a narrative introduction explains the 
theory, next step-by-step procedures are provided, and worked examples are given, and 
finally problem sets are assigned in each chapter, in that order. Material for the worked 
problems and problem sets is situated in realistic behavioral science settings. 

The first stated goal of the course is to familiarize students with the tasks and 
tools of descriptive and inferential statistics so that when they take a subsequent research 
methods course, they can assess others' use of statistics and begin to learn to design their 
own studies. It is not expected that they do these things in PSYC/STAT 104; the 
problems posed in the text and in problem-working sessions have all been worked 
through to isolation of variables. There are no open-ended questions or non-routine 
problems. The text contains no projects to give students experience with the process of 
conceptualizing a hypothesis through data-gathering; the assumption is that these will 
come later in the research methods courses. Nevertheless, the department had designed 
MTNITAB computer projects'^ where students analyze given data sets and learn to 
interpret results. Aim also teaches the Research Methods m Psychology (PSYC 220) 
course and she told me that her expectations of how much was retained from 
PSYC/STAT 104 were fairly low. If students have developed a basic idea of the 
rudiments of descriptive and inferential statistics and their differences, she is prepared to 
re-teach other pertinent PSYC/STAT 104 material during PSYC 220 (personal 
communication, September 12, 2000). 



145 

In order to "increase understanding and critical thinking about the statistics that 
the media presents" (PSYC/STAT 104 Syllabus, see Appendix G) Ann raised some 
common misconceptions around statistical ideas and discussed these briefly with the 
class. She took time in Class 1 to introduce such a problem using a misleading 
advertisement. I took this to indicate that she considered discerning misleading statistical 
information as an important theme for the class. In Class 6, Ann distributed an article" 
that claimed, that an increase in excise tax on beer would "lead to" a reduction in the 
gonorrhea rate amongst teenagers, based on a correlational fmding. She pointed out to the 
class the misattribution of a causal relationship, where a possible link was all that could 
be claimed. Lee was the participant who showed the most curiosity about these issues and 
was very eager to spend more class time than was given to explore them 

Pedagogy and Student Responses 

Ann's approach to mathematics teaching cannot be easily categorized. She did not 
demonstrate how to do procedures; instead she employed a student-centered exploratory, 
problem-working approach to mastering them, expecting that students had the capacity to 
do it, with herself as coach. This approach would be considered pedagogically sound 
from a cognitive constructivist point of view. Because of class time limitations and the 
applied statistical focus of the curriculum, a compromise had to be made between 
presenting conceptual links among and within procedures to the whole class and giving 
students the opportunity to struggle with procedures so that they could master them. Ann 
chose the latter alternative but helped students with conceptual questions and difficulties 
on an individual basis during problem-working sessions. 



146 

Lee's experience illustrated the difficulty an under-confident but conceptually 
oriented student may have in a course like PSYC/STAT 104 even when the instructor and 
the mathematics counselor are affirming of a conceptually curious orientation. Ann 
admired Lee's inquisitive approach and her penetrating questions about the statistical 
concepts but at times Lee was not able to articulate her question clearly or there was not 
enough class time to pursue it. Lee's initially sound understanding that correlation cannot 
be assumed to imply causation as well as her sense of Ann's ability to provide a secure 
mathematics base were each undermined by her perception of Aim's and the text's 
position. ^' 

Ann's non-directive worksheets provided students with in-class experience of 
working through problems on their own^" with her guidance (see Appendix G). This 
process often challenged and even frustrated students. At the same time, each student did 
experience successful completion of at least one problem of each type. I made note to 
discuss in counseling both the appropriateness of their heightened emotions under such 
circumstances and also the pedagogical benefits of this approach. Jamie claimed to be a 
visual learner and said she found the worksheets very helpful, especially the ones with 
the columns, because she felt they complemented her learning style. Mulder also 
preferred to use visual learning approaches and found the worksheets helpful but he used 
them unconventionally and studied by visualizing his successfully worked examples on 
them. 

Students with sound mathematical foundations (e.g., Lee) responded well to the 
challenge of this approach and at times went beyond mere procedure on their own."" 
Students whose mathematical foundations were poor (e.g., Karen and Kelly) found the 



147 

exploratory, problem-working approach difficult and became anxious. Used to having 
procedures demonstrated, Karen'" felt abandoned and helpless when she was expected to 
negotiate such procedures on her own. Both Karen and Kelly complained that Ann had 
not been "thorough" in covering the material before the first test. It may have been the 
absence of familiar solution demonstrations they complained about. I was able to support 
some students as they worked through ihistrations with Aim's exploratory approach (e.g., 
Karen). With this help, they found that they eventually benefited from having to struggle 
to master the procedures on their own. 

Students reacted differently to what appeared to them to be a laissez-faire 
approach to linking the statistical concepts with their underlying mathematical basis and 
to understanding the formulae to number to concept links. All students in the class except 
Lee, Robin, Catherine, and perhaps Pierre were used to following a procedural approach 
to mathematics. Because Ann allowed students to use formula sheets in exams with some 
verbal identifiers and charts, there was a reduced load on memorization of formulae but 
an increased call for understanding differences and similarities among formulae. Because 
the concepts were not uniformly coimected to procedures during class, some students 
found learning new formulae and procedures to be onerous and memory-dependent, 
because they seemed new and different rather than being rooted in previously mastered 
material. 

Even though the more procedural learners were used to this experience, the fact 
that they depended on their memory of dimly understood, individually mastered 
procedures kept them vulnerable. Generally they were without the mathematical tools for 
monitoring and checking and this kept them anxious and dependent on factors they often 



148 

felt were beyond their control. Students like Karen and Brian tended to approach each 
inferential test as if it required an entirely new procedure — another observation I used to 
inform my mathematics counseling. 

Conceptually oriented students found Ann's indirect approach to the conceptual 
linking difficult in some ways, especially if their confidence in their own ability to 
discover these conceptual links was shaky. Lee was the most vocal of the participants 
about her difficulty with this approach but she struggled to make connections herself — 
she attended study groups and met with me to explore and seek answers to her questions. 
Lee spent little time doing homework on her own (20 minutes a week, see Appendix H, 
Table HI) and expressed high anxiety. This may have been related to her difficulty in 
acquiring a secure conceptual base more or less by herself 

Pierre used an opposite tactic to try to gain a conceptual understanding of the 
material. He spent many hours (17 per week at least, see Appendix H, Table HI) studying 
the text and other materials he got from Ann and meeting frequently with Aim and me. 
This broad-based, over-inclusive approach was done at the expense of mastering the 
procedures to be tested and, therefore, at the expense of earning a good grade (at least 
through Exam #3). 

A challenge for me in counseling was to support students' strategic pursuit of the 
conceptual links that were not provided in class and to help them embrace rather than 
resist the real benefits Ann's approach afforded them in mastering the material. 
Mathematical and Statistical Challenges 

Aim was confident in her grasp of the statistical concepts, but she was less 
confident of her grasp of the links between the statistical concepts and the mathematics 



149 

used to explore them. The mathematical challenge of this course Ues principally in being 
able to understand, decode, and link data, and information about data, with appropriate 
symbols or formulae, and in being able to adapt and apply mathematical understandings 
to an unfamiliar problem situation. For example, the order of operations agreement 
requires that to compute DX^ one must square all the Xs first before one adds them (i.e., 
work exponents before multiplication or division, which is, in turn, worked before 
addition or subtraction), whereas for (SX)^ one must add the Xs first and then square the 
result because of the parentheses that require attention to operations inside before doing 
anything else (essentially allowing one to cut in line). In algebra an equivalent situation 
might look like X^ + Y^ + Z^ where X = 2, Y = -3 and Z = 1 compared with (X + Y+ Zf 
when X = 2, Y = -3 and Z = 1. If order of operations is not made explicit, students often 
make errors that they would not if they were simply doing algebra. Because the text does 
not make explicit the equivalencies despite the unfamiliar look, I realized that I should 
include that discussion in counseling sessions. 
Statistics and the Use of Already Derived Formulae 

This course required very little" manipulation of algebraic variables as is typical 
in a non-calculus based introductory statistics courses; there was a heavy emphasis on the 
use of already derived formulae. A conceptual approach to instruction might involve 
explormg the forms of these formulae in relation to their derivations and uses. Formulae 
such as the one used to find the percentile rank of a score (see Class 2) comprise all the 
steps of a multi-step process in one formula; this could be too complex for algebraically 
challenged students because of the intricate interactions among letter symbols and 
operations. I believed students might understand the finalized complex formulae if they 



150 



explored and mastered the process using estimation, proportional reasoning and 
dimensional analysis. In Study Group 1, that is how we approached it (see below). We 
extended beyond using a formula for a percentile rank to fmd its corresponding score 
using the text's step by step approach, and it seemed that the work in Study Group 1 did 
complement the text and class work and forged conceptual connections for some of the 
students. 

Early in the course Aim showed a preference for using an empirical (rather than 
computational) process and formula for finding the standard deviation. She said she 
wanted to help students develop a sense of how and why the formula was derived and is 
used. She had students work the procedure in Class 3, but time constraints and most 
students' procedural orientation led to a predominantly procedural focus for most 
students. In Classes 6 and 7 the concept of deviations and squared deviations from the 
mean reappeared (now in the context of two rather than one variable in correlation and 
regression analysis). Now Arm had students use the computational formula rather than 
the empirical one, and did not link the idea to students' prior work on deviations. This 
was perhaps because it was now being applied to two variables and between the 
variables, in two dimensions rather than one. Although the standard deviation concept 
was the same, the uses and interactions may have been more complex than Ann felt the 
students needed or had the time to explore. There were other mathematical themes that 
Ann did not point out to students such as the fact that the fiinction of all the z and t 
statistic formulae is the same."' I resolved to address these strategically in counseling. 
For example to demonstrate the equivalencies of the t and z formulae, I decided to use 



151 

comparative diagrams (for all) and algebra (only with students who had a level 4 
understanding of the variable on the Algebra Test). 
Multiple Uses of Letter Symbols 

The multiple uses of letter symbols seemed to be the cause of much confusion 
even for relatively algebraically confident students (as noted below in my discussion of 
Class 2, Exam #1, and Class 13). These different uses are not usually discussed m 
application classes like this one, yet they are particularly salient in introductory statistics 
courses because of the heavy emphasis on the use of already derived complex formulae. 
Philipp (1992) notes that current teaching practice in algebra does not address these 
different uses of letter symbols explicitly. In introductory statistics courses instructors do 
typically discuss the symbol classifications of random variable (a true variable), 
parameter (constant for a particular population) and statistic (constant for a particular 
sample). What Arm did was identify names and meanings of important letter symbols as 
the text did. She required accurate memorization of these on the tests, giving up to ten 
percent of test grades to symbol identification and meaning. However, she did not discuss 
classes of symbol, nor draw attention to the multiple uses in one formula, or how the 
symbols differed in their uses, and how they were related to the mathematical content and 
each other."" Because Ann did not provide secure base support in this for students to 
explore and develop these cormections, I took it to be part of my complementary teacher- 
parent role to do so in mathematics counseling sessions. 

Group Learning 

From the first class, Ann provided opportunities for students to work together 
both mformally and formally. The ways students did or did not take advantage of these 



152 

Opportunities or form alliances independent of Ann provided important data about their 
mathematics relational patterns that informed counseling. Although there was no effort to 
organize students to work in groups in class, a paired getting-to-know-you interview in 
Class 1 and pairing up to work on and present the MINITAB computer modules at the 
second to last class presented opportunities for students to form study alliances. Aim 
encouraged students to use the class contact sheet with e-mail addresses and phone 
numbers to contact one another. The only pair of computer project partners to work 
together on other aspects of the course was the Lee-Mitch pair who had known each other 
before the class began. 

Whether students worked together during the problem- working portion of each 
class seemed to depend on where and beside whom they were sitting and on their 
established interpersonal relational patterns. Lee (a social learner'^") initiated and 
maintained contact with Mitch; Robin with Brad (both older and nurses) worked together. 
They formed pairs that fairly consistently sat together and worked together on the 
problems. Lee and Mitch were also MINITAB computer project partners. Mulder (who 
was also a social learner) would work with whoever sat beside him unless it was a loner 
who would not engage. Autumn, Karen, Catherine, (and Mitch if he weren't with Lee) 
Jamie, and Pierre were all loners, rarely working with others, especially other students. 
Autumn, Karen, and Catherine (and Mitch) seemed to be loners by choice (voluntary 
loners), but Jamie and Pierre worked alone more because of constraints they seemed to 
feel precluded choice (involuntary loners). Jamie and Pierre appeared to want to be more 
involved with others. 



153 

During problem-working sessions, in particular, how these distinctions played out 
in class was affected by student seating choices and apparently affected the relative value 
students received. For example, when Mulder, a social learner who found the lectures 
difficult to process and relied on the problem-working session, sat between loners 
Autumn and Pierre, he worked on his own (Class 3). That he did poorly when that 
material was examined in the first exam may have been related. These distinctions also 
seemed to affect the amount of support students received from Ann during problem- 
working sessions. For example, because Jamie rarely used body language that would 
invite Ann's intervention, such as moving to allow Aim to see her work as she went by, 
Ann checked her work and offered her assistance less than she did the other students in 
the class (cf video-recording of Class 16, archived). Because I observed how students 
related (or not) with Ann (and me) in the classroom and I discussed with the student in 
counseling, what that revealed about their teacher attachment patterns, some participants 
were able to recognize and modify such behaviors they now recognized as 
counterproductive. 

My analysis of student seating choices indicates that, contrary to Aim's perception 
of randomness, most students were quite consistent in their seating choices and that my 
choices did not appear to influence theirs. My seating choice did affect the level of 
interactivity of my immediate neighbors during problem- working sessions, however, 
especially voluntary loners like Karen who would not work with her peers but would 
work with me. Seating choices of those who were not loners did seem to be related to and 
affected the level of collaboration during the problem-working sessions (see Appendix 
F). 



154 

Classroom Emotional Climate 

The PSYCH/STAT 104 class had a generally positive emotional climate. The 
course was taught in a manner that had the potential to develop, maintam, and repair 
attachment relationships between teachers and students and between students and 
mathematics. Ann provided the elements of such an environment, but that did not mean 
that each student was aware of it nor received it as a benefit. 

In Ann's course, Jamie and Karen, for example, each of whom came to the class 
with a history of mathematics classroom experiences that had negatively affected them, 
did not initially perceive Arm's classroom as safe for them and could not benefit from her 
positive offerings. In counseling sessions, I saw an aspect of my role as helping them 
investigate whether this classroom climate might be different and even positive for them. 

Dimensions of a positive emotional climate emerged as (I) the creation and 

maintenance of a positive interpersonal relational climate and (2) the creation and 

maintenance of a positive classroom mathematics climate. 

Creation and Maintenance of a Positive Interpersonal 
Relational Climate 

There were three crucial elements to the positive interpersonal relational climate 
that Ann created in the class: herself as a secure teacher base, the classroom as a secure 
base, and fairness in testing. 

The teacher as secure base. Ann provided herself as, a good-enough, emotionally 
secure base for her students so that they can find acceptance and reassurance when they 
are uncertain, as well as the courage to move out to explore without fear of censure for 
going away or for making errors. Arm set the scene in the first class by self-disclosing; 
she described her own struggles with statistics learning and also how she managed to 



155 

overcome her uncertainties."'^ Ann did not hesitate to consult with me in class if she were 
uncertain on the mathematical material, modeling an open exploratory approach that did 
not require students or even teachers to have perfect understanding. 1, too, openly 
expressed my puzzlements.'" 

Another feature of Ann's approach was that she did not call on individuals for 
responses to questions during the lecture discussion. I drew Jamie's attention to this 
during counseling and she was then able to acknowledge to herself that, in this class at 
least, she was safe. She came to realize that she did not have to worry that the instructor 
might call on her.'™' This recognition freed her to relax and even to ask a question of Arm 
in class. 

Although Ann made herself available to meet with students and to help them with 
the course material (because she believed mastery itself would allay anxiety) she did not 
believe it appropriate for her to get involved directly with students' emotional problems 
with mathematics or the class. She neither invited nor required student disclosures. 

The classroom as a secure base. Aim modeled and monitored interpersonal 
classroom behaviors to ensure that all students were safe. The way Aim deak with 
incorrect or half-correct responses during the lecture sessions set the tone. She considered 
the response, found what was reasonable in it, responded, and moved on respectfully. 
Whether a student perceived this positively depended on his""" already established 
interactional patterns. Karen gave an incorrect response to a question during the first class 
but despite Ann's respectful response, subsequently responded only to questions 
requiring a non-mathematical response. 



156 

There were no incidences of student to student disrespect during the course. Lee 
did object to the fact that (in line with department policy and for statistics education 
purposes) Ann distributed a histogram of exam scores after each exam. Because it was 
relatively easy to identify each person's grade given the small class size, Lee felt that this 
was not respectful to students who did not wish to reveal their grades. 

Fairness in testing. Aim seemed to make it a priority to be explicit and fair but 
Karen and Kelly, for example, did not see that. This became a focus in counseling 
because while they were extemaUzing their difficulties and scape-goating the instructor 
they were not taking the control they needed to negotiate the course. 

Before each test Aim was careful to give a study handout with a Ust of the 
symbols that would be tested and specific homework problems from the text. She also 
handed out solutions to even-numbered problems from the teacher's edition of the text 
(for an example, see Appendix G). More importantly she made sure to teach everything 
that she tested; in particular she made certain that each student completed each type of 
problem correctly in class. Ann allowed unlimited time as well as the use of a formula 
sheet on tests. She provided helpful organizers, including the Ust of six steps of 
hypothesis testing, so that students could incorporate this into their formula sheet (see 
Appendix G). 

If a scheduled test time was inconvenient, students could take exams early, though 
not after the scheduled time. Ann's optional comprehensive final could also replace one 
missed exam and could be used to replace a poor exam grade during the course.'"'" The 
relative proportions of the grade allotted (Ninety percent of the grade was earned from 
exams and 10% from computer analysis projects.) seemed to accurately parallel the effort 



157 

and emphasis required in the course. The heavy weighting of exams may have 
contributed to the class' collective mathematics testing anxiety remaining considerable: It 
changed ixom 2.9 (on a scale of 1 : not at all frightened, to 5: very frightened), to 3.0 on 
that scale. Only two individuals' testing anxiety levels fell substantially during the 
course while three individuals' anxiety rose substantially and the others' remained 
substantially the same (see Appendix C for the surveys and Appendix H, Table H3 for 
student changes). One whose anxiety abated somewhat still expressed elevated anxiety 
(3.6); in fact of the 9 students remaining in the class, 7 expressed anxiety levels of 3 or 
above. 
The Creation and Maintenance of a Positive Mathematics Climate 

Aim provided herself and the classroom as a secure relational base, but even that 
was not enough to create a good-enough mathematics classroom climate. Her attitudes 
towards her students' ability and potential to learn mathematics and the way she taught 
mathematics and supported students were also essential. In particular, her belief in every 
student's potential to master the statistics (given adequate support) and her promoting the 
authority of the mathematics over her own authority were key. This was evident in Ann's 
willingness to acknowledge her own uncertainties about the mathematics and refer to 
others (me in this case) who could not only help her understanding but also was there to 
support her students mathematically. 

In Ann's assessments of students' likelihood to do well in this class, her central 
consideration was whether their mathematics background was adequate and whether they 
would apply themselves sufficiently to succeed. She made no trait judgments that might 
have locked students mto doing poorly because she expected it. She did not believe that 



158 

some people could do mathematics and others not. Ann's expectations seemed to be 
influenced by students' classroom behaviors"'" and by a constellation of age, gender, and 
particular major. For example, she (and I, initially) expected Robin, an older (in her 40s) 
nursing student who often appeared flustered and confused in class, to have trouble and 
perhaps do poorly. On the other hand Brad whose classroom behavior was confident and 
apparently relaxed, Ann expected to do well despite his being an older nursing major 
(Interview 2). In each case Ann's expectations were challenged by the student's 
achievements — Robin did well while Brad struggled to get C~s. However, I never 
observed that Aim's expectations affected how she related to or graded a student. 

This apphed course was taught by psychology rather than mathematics faculty, 
and because of that an important complementary role emerged for the mathematics 
counselor. When uncertain about the mathematical bases for the statistics, as noted 
above. Aim was very open about drawing fi-om my mathematical expertise in class. Her 
pedagogical approach, especially her use of problem- working sessions, reinforced the 
statistics/mathematics as authority rather than the instructor. Mathematics counseling was 
pivotal in complementing and supporting Ann's mathematics teaching because of varied 
student comfort with and responses to it at least initially. 

An important part of students' developing a sense of mathematical safety was the 
support offered outside the classroom, especially for those whose low confidence made it 
difficult for them to study and practice on their own. Ann repeatedly offered extra time 
and help to students. Because I was so available and she was at State University in 
another capacity several days a week, most students saw me more than her outside class 
but they were well aware of her openness to helping them. She stayed after class to help 



159 

anyone who came and helped out in several study groups before tests. All six students 
who filled in the Class-Link Evaluation at the end of the class responded positively to my 
contributions as a class-link tutor but only Lee, felt that Ann had relied too much on me 
to give support to students (see Appendix C for the form; student responses are archived). 
CHRONOLOGY OF PSYC/STAT 104, SUMMER 2000 

The class chronology underscores the significance of understanding students' 
mathematics relationality within the whole class system. To describe class process, 
interactions, and student outcomes, I will describe in detail the first three weeks of the 
course through the first test and I will discuss how this was the first of several cycles of 
class, study group meetings, individual counseling sessions, that culminated in an exam. I 
will then sketch key events that occurred during the remainder of the course (see 
Appendix I for a complete calendar of events for the class). 

The First Cycle through Exam #1 
Class 1 

The first class consisted of introductions along with an overview of the syllabus, 
course schedule, and assessment procedures and an interactive lecture on the first chapter 
of the text. All of the 13 students Aim expected were there except for Mitch who would 
be at the next class. 

Aim began to establish the relational climate that accepted struggle and 
acknowledged the importance of collaboration and mutual support by self-disclosing her 
own statistics anxiety (see endnote xiv), by asking the class to pair off, interview each 
other, and mtroduce his interviewee to the class,'™ and by organizing an exchange of e- 
mail addresses and phone numbers. 



160 

After break, Ann used the overhead projector to show an advertisement for paper 
towels that used misleading graphics and numbers to compare with its rival. Lee was 
quick to respond accurately to Ann's questions about it. During the lecture, Ann directed 
her questions to the class as a whole, not to individuals; if there was no response within 
two or three seconds, she answered them herself Ann asked and then explained what 
statistics was, using the defmition given in the text: "A way of organizing, summarizing, 
and understanding data." Data is "information collected and generally understood at a 
numerical level." All of the students wrote the defmitions in their notes. Next the class 
discussed the scientific method and Mulder responded by referring to his research project 
on caterpillar aggression. 

The classroom interactions proceeded in the following pattern: Ann presented a 
concept, she asked a question about it of the whole class, a student or group of students 
responded (or Aim when there was no quick student response). Aim responded to student 
responses, and then cycle began agam. Ellen, Robin, Mulder, and Brad responded during 
this discussion. Robin seemed to have some concepts confused but Mulder, Brad, and 
Ellen appeared to have a good grasp of the big ideas. Karen responded to a question 
incorrectly. Ann dealt with this by respectfully considering Karen's answer, correcting it, 
and moving on. Neither Catherine nor Autumn offered any responses but they appeared 
to be actively and knowledgeably engaged in observing the interactions and they were 
taking notes. Pierre also did not offer any reactions but he was working at his notes and 
attending to the interchange. Jamie alone did not seem to be involved. She kept her eyes 
lowered, not making any eye contact. She did take notes but at times I wondered if she 



161 

were asleep. I did not observe any interactions between students during this class other 
than the paired mterviews. 

Class 2 

The plan for the class was to cover chapters 2 and 3 focused on basic 
measurement concepts and frequency distributions including fmding percentile ranks and 
percentile points. 

Before beginning the lecture, Ann introduced me as researcher and academic 
supporter (class-link) for the class. I invited students to a weekly study group before each 
Wednesdays' class in room 207, where the class met. Mitch had joined the class but Ellen 
was absent. 

In the first half, the class worked on a teacher-directed, lecture-guided data sorting 
exercises, classifying and sorting different types of data according to measurement 
scale™ and fmding the median, mode, and mean of a set of ratio data (time in seconds for 
20 rats to run a maze). Next came sortmg data into a grouped frequency distribution. Aim 
provided worksheets for these exercises and she used the overhead projector to gather 
class responses. The problem-working interactions were aknost exclusively between 
mdividual students and Ann, rarely among students. 

Just before break, according to prior agreement with Ann, I administered the 
surveys I had prepared — the Mathematics Beliefs Survey, the Mathematics Feelings 
Survey, and a short mathematics background survey (all class surveys are in Appendix C) 
and invited volunteers for the individual mathematics counseling sessions. Nine students 
of the twelve present volunteered to be participants. 



162 

The focus of the second part of the class was to learn to use the grouped 
frequency distribution to find a score given a percentile rank. For example, we had to find 
the number of seconds it took a rat to complete the maze given that it was at the 40"^ 
percentile rank in relation to the other rats' scores. Aim commented, "Students say this is 
the hardest math in the course," but assured us that "the math gets easier; the concepts get 
harder." Ann lecture-guided us through the steps delineated in the text as each individual 
worked on the problem and reported his findings. 

No one in the class, with the possible exception of repeating students, had seen 
this procedure for fmding a score in grouped data, given the percentile rank. During the 
procedure I felt lost; I did not have a sense of the end from the beginning, the rationale 
for each step, nor any visual connection with the data — a very uncomfortable experience 
for a conceptual learner like me. I made a note to explore the logical and visual 
connections with students in study group and in counseling. 

Perhaps more importantly for students was the fact that in the formula there were 
six unfamiliar letter symbols, five of them with subscripts. I suspected that students might 
find this procedure difficult on the test the following week. 

By the end of class time the direct process had not yet been tried — fmding the 
percentile rank for a given score. We were assigned this as homework — ^to find the 
percentile rank of the rat that took 82 seconds to do the maze. A formula was given in 
the text but there was no step by step procedure. 

Study Group 1 

The group formed before class at Riverside Center; Brad, Jamie, and Lee were 
there, and Pierre came in a little after we began. The group gathered at the end of the 



163 

table near the chalkboard. The first exam was scheduled for the Monday following, on 
chapters 1 through 5 of the text. Students' pressing concern was solving the problem that 
had been assigned at the end of Class 2. 

I wanted the study group to be a setting where the students did the work while I 
coached. I expected that the students might want help with strategic planning, knowing 
what and what not to concentrate on for the test. Although I had tutored students taking 
PS YC/STAT 1 04 in the semester before this, the instructor was different so like the 
students, I was uncertain how we would be tested. Our natural questions were: Would 
Aim examine students on what had been covered in class or on any concepts included in 
the first five chapters? Would the problems be straightforward or tricky? 

Even before the exam, we had some reassuring evidence that Aim's test would be 
fair. She had handed out a study guide for the exam that included instructions on what 
could and what could not be included on a student's formula sheet and a list of 13 
symbols, including E, a, ji, P5o,and z, whose definitions were to be examined. She had 
included an example of an acceptable defmition. I had observed Ann careflilly checking 
her lecture notes, apparently to ensure that she had covered everything. She assigned only 
certain homework problems from the text and handed out solutions to any even numbered 
ones whose solutions were not in the back of the text. With this evidence I speculated 
with the students at study group that Ann would test only on what had been covered. 

There had been no opportunity for students to explore the derivation of the 
formulae for percentile point and rank using proportional geometric reasoning, so I 
thought the study group could try that. I put the grouped data on the board. Brad took the 
chalk; I coached pointing to and drawing in the geometric proportions on the board; he 



164 

had taken this class before and said that all he wanted was to know how. He did not want 
a deep explanation. 

As students worked, Lee found an anomaly in the formula to find percentile rank 
of a score, namely /, in which the / that is the subscript of/ in the numerator locates the 
interval in which the X score in focus is found (different for different values of X), 
whereas the / in the denominator stands for the size of each class interval which is 
constant for the distribution. The other students in the study group were struggling to sort 
out the other symbols so were not engaged in Lee's and my discussion of this point. She 
decided to use different symbols to keep them straight: an upper case / for the constant 
size of the class interval and the lower case / for the subscript that indicated whatever 
interval we were interested in. I was impressed with Lee's interest in and good analysis of 
this use of letter symbols and showed my enthusiasm for her approach. 

Jamie was actively following the discussion of the process but was not saying 
anything, so I asked for her answer at one point. Although her face turned red, she 
answered correctly. I hesitated to direct many more questions to her because of what 
seemed to me to be her obvious discomfort, though I did some further questioning. 
After completing the homework problem we tried one going in the opposite direction 
and finally stopped as other students came into the room for class. 

Class 3 

Ann handed out written instructions on how to construct a formula sheet for the 
test next class and a reminder of the symbols we needed to be able to define by giving 
both a description and a statistical meaning. On the board, she went over the homework 



165 

question on finding the percentile rank of the time of the rat that ran the maze. Karen was 
the only student apart from Catherine and those in the study group who had done it. 

Ann went on to chapter 4 on measures of central tendency and variability. 
Following a lecture discussion of mean, mode, and median. Aim handed out a worksheet 
to teach the process of finding standard deviation from the mean using the rat maze time 
scores in a column labeled X, and two blank columns labeled to facilitate correct 
interpretation and use of the standard deviation formula.™' There was a brief discussion 
of the different formulae for standard deviations of samples versus populations and Ann 
called the Greek symbol for population standard deviation (o), "omega." Mulder 
corrected her telling her it was called "sigma." Ann accepted the correction but 
erroneously appUed it to s, the symbol for sample standard deviation, and continued to 
call cj, "omega." I felt awkward as she continued and, perhaps somewhat 
inappropriately, discussed the problem with Lee who was sitting beside me. She was 
becoming confiised especially when Ann began calling s sigma. 

This was a very real dilemma. I knew that letter symbol classification caused 
students difficulty and now the instructor was confusing their names and was also mixing 
up the Greek versus Roman letter symbol categorization. A relatively consistent 
convention in statistical symboUzation is to assign Greek letters as symbols for 
population parameters, corresponding to the first sound of the item labeled, for example 
|x (Greek lowercase "mu") for population mean, and a (Greek lower case "sigma") for 
population standard deviation. Roman letters are used to symbolize sample statistics, for 
example, X is used for sample mean and 5 for sample standard deviation. In my 
experience, explaining this convention to students struggling with many unfamiliar 



166 

symbols had helped them considerably. 1 felt compelled to clear up Arm's confusion with 
her so the students would not be confused but on the other hand, 1 was very aware of the 
delicacy of trying to balance my multiple roles of participant as a student, class- link tutor, 
researcher, and colleague m relation to Ann's instructor role, and I was not sure how Ann 
would react to my introducing this concern. 

When I did address it with Ann privately after class her reaction reaffirmed my 
prior assessment of her healthy self-reliance (cf Bowlby, 1982) and of an appropriate 
way for a class-link tutor to approach such an issue with a self-reliant instructor. Rather 
than my telling her of the error, I pointed it out by using the text as the authority. 
Although a Uttle embarrassed, Aim reacted positively to my addressing it in private and 
to now knowing the correct designation for a and s. The way 1 handled the incident 
seemed to contribute to Aim's confidence in my expertise because she encouraged me to 
contribute it in the fiiture. At the next opportunity Ann addressed the confusion directly 
with the class. I was then able to emphasize the Greek-Roman letter distinction for 
population versus sample symbols with individuals without involving them in an 
apparent conflict between believing what the instructor taught or what I told them. Kelly 
was the only student in the class who remained confused labeling a "omega" in Exam #1. 

Ann had filled in the bottom total row of the X - mean column on the worksheet 
with the mathematical statement, S(X - mean) = 0, that is, the sum of the deviations of 
scores from the mean is always zero no matter what scores are analyzed. This indicated 
to me that Ann wanted the students to attend to that concept particularly but she didn't 
stress it in class. Its mathematical inevitability, its universal application throughout 
statistics, and its usefulness for checking ones' mathematical processes seemed so 



167 

important, however, that I took mental note to point some of this out to students in 
individual sessions. 

At this point in my classroom observations I was using the dialogue and the class 
plan as recording devices (see Appendix C) for interactions between instructor and 
students during lecture discussions. I was not yet systematically recording interactions 
among students during problem-working sessions. Ann circled the room helping 
individuals and there seemed to be little interaction among students. Later I reaUzed that 
students, who in later classes worked together, were not sitting together in this class, 
except for Robin and Brad. Mulder who later would work with anyone who was willing 
was between Autumn, who never worked with a fellow student in problem-working 
sessions, and Pierre who rarely did. This setup definitely did not lend itself to 
conversation. 

After break Ann covered chapter 5 (T/ze Normal Curve and Standard Scores) in 
lecture discussion using the rat maze mean times and standard deviation computed before 
break to compute a transformed z score on the board. She also demonstrated use of the 
standard normal tables to find the probability of obtaining a particular time score or less. 

At the end of the class students who had formed pairs for the MINITAB computer 
module project were given their projects. Lee had already arranged to partner with Mitch. 
Otherwise, women sitting beside or near each other in the previous class seemed to have 
already paired off when Aim made the announcement. (Kelly worked with Autumn and 
Catherine with Karen.) Men who were sitting beside each other in class 3 paired off then: 
Mulder partnered with Pierre and Brian with Floyd. Robin was left out — she seemed 
flustered by the situation and initially said to me that she would just do the project on her 



168 

own. When I told her that I understood that it was a requirement to partner with someone, 
she agreed to partner with Jamie who did not yet have one. 

Individual Sessions 

The first test was to be held at the next class. I had individual mathematics 
counseling meetings with Kelly on the Thursday and Karen on the Monday just before 
the test. Both were very anxious, each seemed to have some fundamental arithmetic 
confusion with decimals and percents, and both found the large amount of material to be 
covered in the test overwhehning. 

I had also been involved with Pierre who had to take his first test early because he 
would be away when it was scheduled. He was going to take it in the library (located 
across the hall from the Learning Assistance Center at the Greenville campus) on the 
Thursday (June 8). He took the day off work so that he could study. I was expecting that 
he would come to the Learning Assistance Center. At one point in the mid-afternoon 
when there was a lull in Learning Assistance Center activities, I went looking for him and 
found him in an otherwise empty conference room. He said that he had looked into the 
Learning Assistance Center earlier adding, "You were busy." I encouraged him to come 
in next time and begin working so I could help when I was free. There were several 
similar situations in the following two weeks until we finally managed to meet for his 
first mathematics counseling session towards the end of the fourth week of class. 

On Monday, the day of the first test, Karen and Kelly came to drop-in 
mathematics tutoring at the Learning Assistance Center. Karen came only briefly before 
her mathematics counseling appointment at 4:00 p.m. Kelly seemed so overwhelmed that 
she was seriously considering dropping the class. I felt as if she was trying to get me to 



169 

tell her if she should. I told her that she had to make that decision but that perhaps taking 
the test might help her decide. She decided to meet with Ann to discuss her options and 
perhaps get her to make the decision for her. Kelly's mother had already called Ann and 
me to discuss Kelly and we had independently encouraged her to let Kelly work it out. 

Exam #1 

The first test was held in a room across the hall from where the class usually met. 
The exam room had individual seats with attached small right-handed desks all facing the 
front of the room. I expected a rather high level of anxiety before this first test. Karen had 
expressed some of this, perhaps seeing Ann as the cause, but students always face 
unknowns that cause anxiety in all courses even when the instructor makes an effort to 
prepare them, as it seemed Ann had. 

On the surveys I had administered in Class 2, the class average level of 
Mathematics Testing Anxiety of 2.9 on a scale of 1 (not at all scared) through 5 (very 
much) with a range of 1 .5 through 4. 1 indicated a higher level of anxiety than for either 
Number (2.2) or Abstraction (2.8) and high™'" if compared with Suinn's (1972) norms on 
the Mathematics Anxiety Rating Scale (MARS) from which the items were drawn. 
Jamie's reported testing anxiety level was the highest in the class at 4. 1 and Karen and 
Kelly's were almost as high at 3.6 and 3.5 respectively (see chapter 6 and Appendices H, 
K, and L). In counseling, we explored relationships among participants' testing anxiety 
scores and factors such as their past experiences in mathematics exams, their preparation 
for the exam, and their perceived ability to achieve on the current exams. 

Everyone was in the classroom by 6:00 p.m. except Kelly. Kelly was considering 
dropping the course but Arm had just persuaded her to try the first exam so she came in a 



170 

few minutes late. Each student had to hand his formula sheet'"'^ to Ann and pick up part 
I, the part that Ann called conceptual consisting of multiple-choice questions and symbol 
identification questions. Once a student completed part I, he took it to Arm who was 
sitting at the table in the front of the room, and picked up the computational part of the 
exam with his formula sheet that Aim had checked to make sure it met her criteria. 

Students took between 20 and 30 minutes to complete part I of the test. Part II 
consisted of 14 questions requiring various descriptive statistical analyses of small sets of 
data, all from chapters 2 through 5 of the text. All required procedures had been covered 
in class. Students had had the opportunity to struggle through all the procedures using 
worksheets in class except for the z-score questions. Those had been covered in Class 3 
on the board by lecture discussion only. 

Ann had agreed that I could ask students to complete Joan Garfield's (1998) 
Statistical Reasoning Assessment (SRA)™' after they completed their tests. I intended to 
use this as a pre and posttest to gauge changes in students' statistical conceptions. As 
each student finished her test she took the SRA. Autunm had arranged her first individual 
session for immediately following the test so when she finished the SRA (around 7:20 
p.m.) we went across the hall to our usual classroom and began. Autumn told me that she 
thought she had done well on the test, that she had done quite a bit of statistics before, 
and that she didn't think she needed any help on the current course. As the other students 
completed their SRAs they came in and gave them to me. Catherine pronounced it "very 
hard" and seemed anxious about it. 



171 

Results of Exam #1 

The students' test results are shown in Figure 5.2. My first impressions gathered 
in class, in the study group, and from individual contacts led to my being surprised by 
Jamie's high score because of her expressed anxiety, by Robin's high score because of 
her apparent confusion in class, and by Lee's relatively low score because of her 
insightfol conceptual approach at study group. (On the other hand she had e-mailed me 
the night before the test expressing considerable testing anxiety and asking for my help.) 
Mulder's low score also surprised me, particularly his score of 22 out of 40 on the 
conceptual part (55%), the lowest in the class, because he had been outgoing and 
articulate in class and seemed to have background in the use of statistics in research. I 
found Brad's relatively low score was both surprising and not. On the one hand his 
confident demeanor and class participation bode well for a high score but on the other 
hand, his professed procedural approach in study group with the "only wanting to know 
how" and his admission that he was repeating the class did not bode well. 
Error Analysis of Class Performance on Part II (computational) 

I expected students to have the most difficulty on the question asking, "What 
score is at the 50* percentile point?" not only because of the complex nature of the 
formula, the multiple uses of different types of letter symbols and the largely procedural 
way this was approached in the text and in class, but perhaps even more because it was an 
inverse mathematical procedure requiring students to begin in the right hand cumulative 
percent column that they had to create themselves and proceed left to identify the score 
corresponding to the given percentile. My expectation proved to be well founded. 



172 

Only two students (Jamie and Catherine) did this problem correctly. The other 
nine on whom 1 have data (I do not have Mulder's Part II of Test #1) lost at least half of 
the points given for this question, and more students made errors on this than on any 
other question on the test. Three of these made the inverse error — ^that is, they began with 



0) 

(D 

I- 

C 

o 

O 
U 
(0 



Individual Scores on PSYC/STAT 104 

Test#1 



120 -1 
A c\r\ 




\ UU 




1 — 1 


1 ■ 


■ '"f^ 


ou 


^^ I- 


1 g 


-■- 


■ ' 


I I 1 


40 

20 
n 




- 


■ ' 


u 


1 


2 


3 A 


\ 5 


6 


7 £ 


\ 9 1 


11 


12 


□ Concept 


28 


28 


28 2 


2 32 


34 


30 2 


6 38 3 


8 38 


40 


■ Symbols 


4 


3 


7 "i 


r 6 


5 


8 { 


J 7 ( 


J 7 


8 


■ Compute 


10 


28 


27 3 


4 30 


33 


38 4 


4 41 4 


3 50 


52 



STUDENT 



1 2 



9 10 11 12 



Test Totals: | 42 | 59 1 62 1 63 1 68 I 72 i 76 | 78 | 86 |89 |95 1 100 



Figure 5. 2. Individual's scores on Test #1, with each student's score broken down into 
his Conceptual multiple-choice Part I) score, out of 40; his Symbol score (on Part I) out 
of 8; and his Computational score (Part II) out of 52, total possible 100. 
Note: the X-axis numbers refer to individual students as follows: 1: Floyd; 2: Kelly; 3: 
Karen; 4: Mulder; 5: Pierre; 6: Brad; 7: Lee; 8: Mitch; 9: Autumn; 10: Robin; 11: Jamie; 
and 12: Catherine. 



173 

a score of 50 instead of a percentile rank of 50, but all three had not made a cumulative 
percent column from which they should have begun. 

Multiple uses of letter symbols seemed to be the cause of much conftision in this 
question even for algebraically confident students. Students made the most errors 
identifying the correct number indicated by the letter symbol, especially when it was a 
multi-level symbol such as cum /l that not only requires careful interpretation, but also 
requires a muhi-step approach to computing.'™" Six of the students made errors 
in correctly identifying or computing the cum /l. Of the five who identified cum /l 
correctly, three had attended the study group. 

Jamie and Lee were the only students in the class to fmd and correct their own 
initially wrong substitutions as they were taking the test. 
Grading and Instructor Response to Test #1 

Aim returned the tests to students at the next class with a summary of class results 
m the form of a grouped histogram (see Appendix G) with the mean (74.2%) and 
standard deviation (16.7%) of the scores. Students looked over their tests and then had to 
return them to Ann. Since I was to be working with individuals, I arranged to get their 
tests from Aim if they wished so we could do error analysis during individual sessions. 

Aim seemed to use arithmetical accuracy as the main indicator of correct 
procedure. No more credit was given for correct process with an inaccurate result than for 
incorrect process with an inaccurate result. For example, in question 10, Brad lost 3 of the 
4 points because he had replaced his initially correct cumulative frequencies and percents 
with mcorrect ones and used those with the correct process to find the score given the 
percentile rank. Floyd, Mitch, and Pierre also lost 3 points on that question when they had 



174 

made the much more serious inverse error, treating the given percentile rank as if it were 

a score. With individual counseling participants who focused on grade as the gauge of 

their mathematics ability, I feU an important strategy would be to affirm logical, 

conceptually sound, albeit arithmetically inaccurate work as a more accurate way to 

gauge their understanding and ability than the points they earned (or lost). 

Relationality and Implications for Counseling 
in the Next Cycle 

Before Exam #1 I had met only with Kelly and Karen for individual mathematics 
counseling. After Exam #1,1 met with the others who had signed up at various times and 
by Exam #2 on June 26, 1 had met with them all at least once (see chapter 4, Table 4.2). 
Kelly and Floyd had stopped attending class by Exam #2. All of the individual 
counseling participants who took it except Jamie either maintained or improved their 
scores on Exam #2 (see Table 5.1). Lee and Mulder's improvements were dramatic (up 
two letter grades); Autumn, Karen, and Mitch improved by one letter grade; the others 
made more modest gains. Jamie's decline was as dramatic as was Lee and Mulder's 
improvement — by two letter grades. 

Student MINITAB partners had been working with each other since the computer 
lab class which was held in the computer center at the Greenville campus, but there was 
no evidence in class of these partnerships leading to study alliances. 

The first test marked an important point in the trajectory of the class not only for 
the students but also for me. The test was a key piece in confirming my initial judgment 
that Ann was providing a positive classroom climate where vulnerable students could 
progress with will and strategic support. It also signaled who was vulnerable but in a 
crude way, the grade signifying quite different things for different people. For example. 



175 

Jamie found her 95% quite unexpected, whereas Catherine's 100% was not a surprise at 
all; Kelly reacted to her 59% flustered, casting about for someone to tell her whether to 
continue in the class or not whereas Karen and Mulder with similar grades refused to take 
them as a verdict on their course outcomes. Whether students perceived this positive 
classroom climate or not depended on their past mathematics experiences, the status of 
their mathematics preparation, and the relational patterns in which they were imbedded. 
That was a challenge for counseling: to help students experience the current positive 
relational reality rather than a negative reality from their past. 

In this first cycle through the first test I found that Ann's provision of a positive 
relational climate gave me the space to negotiate a comfortable position for myself as part 
of the class community, and her healthy self-reliance made it possible for her to access 
my support comfortably. As a result, in class and in counseling there had developed an 
easy sense of our working together for the benefit of students. In counseling it seemed 
that I would not have to do damage control for current relational assaults but rather 
develop my role as a mathematics complementer of inevitably underemphasized or 
missing material fi-om class while the student and I explored his particular relational 
challenges. 

The Post Exam #2 through Exam #3 Cycle 

Exam #3 was based in chapters 10 through 14, moving into inferential statistics 
with hypothesis testmg using sampling distributions. Ann confessed to not being really 
clear on sampling distributions and how to explain them to the class. She invited me in 
class to offer my fiirther explanation. I was somewhat nonplussed, not being sure what 
she found confusing or what she felt students did not understand. Her explanation seemed 



176 

clear to me and I said so but I knew that unless students actually created their own 
sampling distribution from a finite population the connections would likely not be clear 
to them — telling is no substitute for experiencing the mathematics. There was no 
opportunity for that now, however, except perhaps with Lee in a study group that only 
she attended. 

Class 13 

The material to be covered in this class on Wednesday, July 12 was chapter 13 on 
the Student's t test for single samples and chapter 14, on the Student's t test for correlated 
and independent groups. The next class would be Exam #3 on all inferential statistics and 
hypothesis testing through this class. 

The first part of the class was a short lecture discussion. Ann explained the use of 
the single sample t test for sampling distributions, comparing and contrasting it with the 
single sample z test covered in the previous class. She briefly mentioned using the sample 
mean to fmd a confidence interval to estimate the mean of sample means of a sampling 
distribution of the population. She discussed power and Type II errors, and then wrote on 
the board a t test confidence interval problem and handed each student a problem sheet 
requiring hypothesis testing using a single sample t test. 

Because of time pressure, Ann did not go beyond helping students work through 
the procedure for finding a confidence interval estimate for a population mean. Students 
then alternated between working alone and with the person beside them (except Jamie, 
Autumn, and Pierre who only worked alone) on the single sample t test problem without 
too much apparent difficulty. 



177 

After break beginning at 7:35 p.m., Ann gave a ten minute overview of two- 
sample t test hypothesis testing comparing the means of two samples. Students were 
directed to work on a worksheet problem, using the text for formulae and as a procedural 
guide. As the students began working on the independent samples t-test, individually 
(Autumn, Jamie, and Pierre) or in pairs (Brad and Robin, Catherine alone and Mulder 
checking with her, Lee and Mitch, and Karen and me), there was an audible reaction to 
the formulae on page 331 (Pagano, 1998). Karen growled, Mulder sputtered in disgust 
and they both proclaimed, "Yuck!" Robin frowned harder than usual and sighed. Mulder 
demanded, "So where's the short version of this?" 

There were a number of potential trouble-spots in the independent samples t 
statistic formula, especially the complex subscript for the estimated standard error. 
Although Ann had not explicitly taught the idea of subscript-as-label, students had 
generally succeeded to this point, apparently by ignoring the subscripts that were 
monomials (e.g., 3 inXsor a/2 in tan)- But they found this new binomial subscript (i.e., 
two terms as in A} - X2) with terms that themselves had subscripts, very confiising or 
rather distracting. Now that they could no longer ignore the subscript, instead of 
understanding that the subscript's function is labeling only, some students tried using it as 
part of a formula, in this case to compute estimated standard error of the difference 
between means {S _ _). Autumn, who always worked alone and rarely asked even 

X, - X: 

Ann a question during problem-working, had done this with the subscript. In an 
unprecedented move, she got up from her seat and came around the table to me because 
she knew that what she had done with this formula was wrong but she was not sure why. 
She had given the subscript a numerical value of (since she knew that the null 



178 

hypothesized population mean difference was zero, that is, i^i - 1^2 = 0), had written the 
population a instead of the sample S and had finished with ct - as her interpretation of 
S_ _ . Instead she should have seen this as the symbol only and computed its value 

X, - X: 

usmg a series of formulae given in the text. This type of error and confusion by even 
confident and high-achieving students led me to predict that symbol and formulae issues 
would cause more difficulty for students on the next exam than anything else (with the 
possible exception of correctly deciding which statistical test was applicable). 

As each student completed the independent samples t test problem Ann checked 
it, and moved him on to the correlated samples / test. This did not cause as much 
consternation as the independent samples / test probably because the formulae are less 
complex and include only monomial subscripts. With each of these unfamiliar statistical 
tests, students were able to use the familiar six-step hypothesis testing procedure protocol 
that Ann had provided them in Class 1 1 — this provided welcome consistency. 

The first students to fmish left at about 8:30 p.m., ten minutes after the class 
officially ended, and others were still there after 9:00 p.m. This was the only class that 
ran over time; as the Exam #3 was to be given at the next class students did not seem to 
react negatively. 

Individual Sessions 

Between Class 1 3 and Exam #3 I had individual meetings with Pierre and Karen 
twice and with Brad and Mulder once. Pierre and I had had two long sessions and I 
realized that he was trying to master all the material in the text, not just what Ann was 
requiring. While this seemed admirable it was leading to his not mastering in sufficient 
detail any of the material, especially the material Ann was requiring, so his grades were 



179 

poor (D* and C~ ) and I expected this exam to be the most challenging thus far. I 
suggested he focus more on what was being covered in the course so he began to work on 
the course specific materials I prepared. Mulder met with me the early on the morning of 
the exam. He said he hadn't done much, if any, study and said he was feeling stressed 
because ha had to work for the rest of the day and would have no fiirther chance to study. 

Karen had met with me the week before the exam and we had worked on the 
Mann- Whitney hypothesis test. On the day of the Exam #3 we met again after she had 
already been at Drop-In mathematics at the Learning Assistance Center at the Greenville 
campus for three hours. Jamie said she would come to Drop-In too but didn't. However 
Ann was offering a special pre-exam study group/drop-in session in our classroom at 
Riverside from 4:00 p.m. to 6:00 p.m. and Jamie was there when Karen and I arrived at 
about 5:00 p.m. from Greenville. Lee had been there since 4:30 p.m. and was workmg 
with Ann. Lee had called me earlier in the day panicked because she had to work all day 
after being ill all weekend. She had hoped to meet with me at 4:00 p.m. but I was already 
scheduled to meet with Karen then. Autuitm and Catherine were also there but each was 
working on her own. Mitch arrived about 5:20 p.m. and seemed unprepared. He had 
been hosting a visitor and said he had not been focused on his work. 

Exam #3 

My anxiety level on behalf of the students was higher for this exam than for any 
of the others. I was especially anxious about how to counsel and tutor those who sought 
help. I was also anxious about those I believed should have sought help and did not. My 
anxiety stemmed not only from the fact that the inferential statistical and sampling 
distribution material being examined was new to most students and was considerably 



180 

more complex than the previous descriptive statistics but also from my undoubtedly 
inappropriate feeling that I could not quite trust the students to take responsibility for 
themselves. I was also concerned with the number of statistical tests and procedures the 
students had to master and wondered with them if they would be expected to identify the 
applicable one from the problem statements on the exam. I e-mailed Ann to this effect 
and she replied that she was not sure; maybe she would identify some and have students 
decide on others. 

To help students prepare for this exam I used my analysis of content of the 
previous exams to develop strategic practice materials. Because the previous two exams 
focused on problems like the ones worked in class, I erased the test labels on each of the 
problems we had worked in class and copied them for individual counseling. I added a 
normal z test problem from the text as we had not done one of those in class. I also 
modified the decision flowchart the text used for choosing the appropriate statistical test 
(Pagano, 1998, Figure 19.1, p. 473) to create a simplified flowchart including only tests 
being examined on this exam, leaving the test name boxes blank for the students to fill m 
(see Appendix J). Ann agreed that this modified flow chart, filled in by the student, 
would be acceptable as a formula sheet to use in the exam. 

The student who took most advantage of these materials was Karen, during her 4 
hours at the Learning Assistance Center on the day of the test. On her second test she had 
made significant errors because of missing work columns on her formula sheet and 
because she had not sufficiently practiced all the problems to be tested. She remedied 
both of these problems for this exam and was rewarded with an almost perfect score on 



the computational section. She was one of only two students whose grades improved 
from Exam #2 to Exam #3, before the extra credit was factored in (see Table 5.1). 

Aim's strategy of having the students work through problems in class without first 
showing them what to do forced each to negotiate the procedures required. The lack of 
systematic group discussion of links among various aspects of the process and of known 
trouble spots, except with individuals, left some vulnerable to memory lapses or 
confusion in the exam unless we addressed these in a counseling session. For example, 
explicit discussion of the direct relationship between the null hypothesis statement and 
the relevant symbols and parts of the / statistic formulae may have prevented the error 
some students made.'^^" In the exam Mitch, Mulder, and Robin (almost one third of the 
class) used their non-zero mean of sample differences D for yi^ instead of zero even 
though they each correctly stated in their null hypothesis statement that there was no 
difference or change in the population scores before and after. They each then had to cast 
about for improbable Ds because they had used theirs for jXa 

Four of the ten students who took Exam #3 made errors in choice of degrees of 
freedom in the independent samples question on the exam and one in the correlated 
samples question. Others made errors negotiating the t table. This procedure was 
introduced to students in Class 13, and its application is complex; it is somewhat different 
for each of the three t tests taught in that class and different in significant ways from the 
famiUar procedures for using the normal z table. Guided questioning in the form of an 
assignment sheet might have helped students become more conscious of these 
differences. In subsequent counseling sessions I noted the importance of walking 
participants through the use of unfamiliar tables. Karen had resolved her initial 



182 

difficulties with the tables during her afternoon of preparation and she had 
simultaneously designed her formula sheet to prompt correct usage. 

Pierre was the only student who misinterpreted the subscript of the standard error 
as part of a formula instead of as a label. Instead of calculating S (=1 .77) using 

X, -X; 

formulae on his sheet, he used X/ - X? not as a label, but as a factor, multiplying it by 5 
to get S( X,- X2), that is, 9.73(43 - 39) or 38.92. The large size of his standard error 
should have given him pause. Because there was no opportunity in class for discussion of 
the expected relative sizes of the statistics, in relation to the mathematical processes 
involved, in mdividual sessions I realized that it was important for me to model and 
encourage students in this type of questioning and checking. 
Grading and Instructor Response to Test #3 

A new feature of Ann's grading emerged with the focus on hypothesis testing. In 
her scheme, a certain number of points, typically 3 or 4, were allocated for the correct 
decision at the end of the process (i.e., whether to reject or fail to reject the null 
hypothesis) and for its meaning in terms of the problem at hand. On one question in 
Exam #3, four students made errors in their calculation of the statistic and found its 
magnitude to be less than the magnitude of the critical value. They therefore logically 
decided to fail to reject the null hypothesis.™"" Ann penalized them the foil amount 
because they made the incorrect decision, even though it was the one demanded by their 
results. I was concerned in individual counseling to affirm students' sound mathematical 
decision- making in a situation like this, and try to allay the negative impact of the lost 
points on their self concept. At the same time as we discussed the validity of the 



183 



instructor's emphasis on the need for the correct decision'°"''and, therefore, the 
importance in subsequent exams of checking the accuracy of one's computations. 
Table 5.1 
Grades Throughout the Course of all Individuals in PSYC/STAT 104, Summer 2000 





Exam 


Exam 


Exam 


Exam 


Minitab 


Minitab 


Final 


Optional 


TOTAL 




#1 


#2 


#3" 


#4 


Module 

1 


Presentation 


Exam 

(#5) 


Comprehensive 

Final 




Percent of TOTAL 
GRADE 


20 


20 


20 


20 


2 


8 


10 


Replace 

lower exam 

grade 


100 


Autumn 4^ 


86% 


96% 


(90+6)% 


95% 


100% 


100% 


100% 




94.6% 


Brad 4 


72% 


69% 


(56+4)% 


72% 


100% 




62% 






Catherine 


100% 


92% 


(91+6)% 


100% 


100% 


100% 


100% 




97.8% 


Ellen 




















Floyd 


42% 


















Jamie 5 


95% 


74% 


(84+6)% 


76% 


100% 


100% 


100% 


71% 


87% 


Karen 5 


62% 


74% 


(85+6)% 


88% 


100% 


100% 


96% 


57% 


83% 


Kelly 3 


59% 


















Lee 6 


76% 


97% 


(83+6)% 


81% 


100% 


100% 


100% 


77% 


88.8% 


Mitch 4 


78% 


87% 


(62+6)% 


82% 


100% 


100% 


92% 




82.2% 


Mulder 5 


63% 


81% 


(76+5) 


91% 


100% 


92% 


94% 




81.96% 


Pierre 8 


68% 


72% 


(60+6)% 


91% 


100% 


92% 


96% 


72% 


79.56% 


Robin 3 


89% 


87% 


(77+6)% 


88% 


100% 


100% 


96% 




89% 



Notes: ' Names of counseling participants are bolded and the number beside their names is the number of 
their counseling sessions. ''Because more than two thirds of the class experienced grade decline, some 
severe, on Exam #3, and more showed a fundamental lack of understanding of the concept of statistical 
power, Ann gave an in-class, open-book assignment worth up to 6 points to be added to the Exam #3 grade. 

Karen and Jamie's scores on Exam #3 showed an improvement of one letter grade 
over their scores on Exam #2. Everyone else except Catherine (whose score remained 
about the same) dropped from one half to two letter grades. Ann was concerned not only 
with the drop in scores but also with the evident lack of understanding of the concept of 
statistical power. In Class 1 5 she assigned an open book extra credit assignment for 6 
points on the topic of statistical power and the factors that influence it (see Table5.1). 



184 

From Exam #3 through the End of the Course 
The ten students remaining in the class were all passing with grades ranging from 
a D"(Brad) through A (Autumn and Catherine) after completing Exam #3. Karen was 
showing steady improvement in grades and Jamie was recovering from her big dip in 
Exam #2. Mulder's score on the multiple-choice conceptual section remained a 
significant problem but he had done quite well on his computation despite his lack of 
preparedness for the exam. Brad seemed quite crushed by his low score and 1 felt the urge 
to "rescue" him from himself, convinced that he was sabotaging his own chances of 
succeeding. Pierre had not followed my advice to focus on course material only and did 
poorly again (a D~ before the extra credit). 

The nine remaining counseling participants continued to meet with me 
individually. Some also attended study group, Lee every week and others if the study 
group was just before an exam. Their course grade progress is shown on Table 5.1. Their 
progress as mathematics learners and course strategists and other changes in their 
mathematics mental health are discussed in chapters 6 and 7. 

END OF COURSE SUMMARIES AND DISCUSSION 
Student-Instructor Interactions during Lecture Discussions 
Analysis of the interactions between the instructor and individual students during 
the lecture discussion portion of class revealed patterns relevant to the emotional climate 
of the classroom and the individual's perception of it. In general Ann asked questions of 
the whole class; she directed questions to individuals only in relation to a prior issue they 
were discussing. At times several students responded together to Arm's whole class 
questions. When students asked questions, some raised a hand to draw Ann's attention 



185 

(e.g., Lee in Class 16); others spoke into a silence or out of puzzlement with what Ann 
had just said (e.g. Karen, Mulder, and Robin each asked Ann to repeat or clarify what she 
had said in several instances). 

There was almost no correlation between the number of students' responses or 
questions and their grade in the class (see Table 5.2). There was, in fact, a small negative 
correlation (r = -.244) between a student's average number of responses or questions and 
final grade (for those who completed the course). 

In addition, apparent accuracy and pertinence of student response was often 
incongruent with grade. These phenomena make it very likely that any judgment of 
student competency based only on class interactions could be quite misleading. 
Table 5.2 

Number of Individual Utterances During Lecture Portion of Classes and Final Grade 



















Class Number 
















1 


2 


3 


(4) 


(5) 


6 


7 


8 


(9) (10) 


11 


12 


13 


(14) (15) 


16 


(17) (18) 


Av. & 
Final 
Grade 


Autumn 





4 


3 






3 





1 













2 







0.82; A 


Brad 


4 


2 


2 






12 


2 


2 










3 


1 


4 




2.9; D 


Catherine 








1 






1 


1 
























0.27; A 


Ellen 


3 
































Floyd 


3 





1 



























1 


Jamie 











































0.0; B* 


Karen 


1 


4 


3 






2 


1 


3 




2 


3 


2 


1 


3 




2.27; B 


Kelly 


1 


3 


2 






1 





2 
















1.33 


Lee 


2 


4 


4 






5 


2 


8 




5 




3 


1 


3 




3.36; A" 


Mitch 




3 


1 










2 




1 


1 





2 


1 




1.38; B" 


Mulder 


6 


3 


3 






3 





3 




7 


5 


1 


1 


2 




3.09; B" 


Pierre 





4 









1 


1 


2 




1 




1 


1 


1 




1; B" 


Robin 


3 


4 


5 






7 


1 


2 




4 


8 


3 


1 


3 




3.73; A" 



186 

Participation alone is clearly not enough. Factors related more closely to classroom 
interaction were student learning style and preferred modality, personality, and previous 
experiences in a mathematics learning environment. 

Brad and Mulder interacted in a way that gave the impression of familiarity with 
and grasp of the material, while Robin gave the opposite impression. Ann's initial 
judgment of Brad's competence was dramatically modified by his poor grades and his 
struggles in the class problem-working sessions. Her initial judgment of Robin's 
incompetence persisted however, even despite her consistently good grades. Both Robin 
and Mulder seemed to have difficulties with auditory processing of lecture material but 
Robin's struggles were clearly discemable in her often puzzled demeanor, her checking 
with Ann to see that she had understood correctly, and in the tentativeness of many of her 
correct responses.™' In contrast, Mulder responded to questions only when he was certam 
of the material; he dealt with his struggles to understand the lecture presented concepts by 
focusing on parts of the lecture and ignoring others or by giving up and working instead 
on the computation with fellow students during problem- working sessions if they were 
willing. After Karen gave an initial response that showed conftision from then on she 
restricted her responses to supplying data (e.g., her beer preference. Class 16; or sports 
data). 

Jamie was the only student who never asked or answered a question during the 
lecture discussions (see Table 5.2). Her shyness was obvious from the beginning and she 
typically kept her head down and eyes lowered. By the end of the course she was raising 
her head and making eye contact but she still did not speak. The strongest students in the 
class grade-wise, Catherine and Autumn, were among the quietest. Catherine (average 



187 

responses 0.27 per class, see Table 5.2 above) had an air of quiet confidence that 
accurately reflected her easy mastery of the material. Autumn (average responses 0.82 
per class) was more responsive during the first half of the course than later, perhaps in 
part because she was more familiar with the material at the beginning and she was careful 
only to respond when she was quite certain. 

All students except Robin seemed to try to restrict their responses to answers they 
felt they knew. For example, Mitch usually held his head stiffly on his hand and would 
respond barely audibly, when he was sure of the answer (see Appendix E, Table E2 for 
the criteria I used to analyze student's utterances during lecture discussions). 
IMPLICATIONS FOR MATHEMATICS COUNSELING 

I found that being in the class, doing the statistics, observing the students, and 
taking the exams provided me with good data to use in order to plan and provide strategic 
tutoring in the statistics/mathematics and the counseling of participants' relational issues. 

Study Groups and Cooperative Learning 

Because the study group was open,'™" attendance fluctuated from one to seven 
students, apparently according to whether there was an exam immediately following (see 
chapter 4, Table 4.2). Because attendance was not consistent, a group working approach 
was difficult to establish. No one in attendance, except perhaps Lee, was oriented towards 
working with peers to investigate a mathematics problem. They each related directly with 
me and seemed to show little mterest in others' responses unless I directed them to 
evaluate those responses. It was easier to have the students work together on a problem 
when there was not an exam immediately following but even then the pressure to master 
the procedures precluded open group explorations. The focus was on working assigned 



188 

problems with my guidance and coaching and with students taking turns in presenting 
solutions to the group. 

The study groups that met before exams were more like drop-in with me (and Aim 
before Exam #3 and Exam #4) moving from individual to individual helping each with 
his particular questions. An exception was the one before Exam #5. That was a round- 
table discussion of problems Ann provided and I supplemented, during which I asked 
individual students in turn (including Jamie) to respond with their solutions. 
Mathematics Counseling and the Classroom — Relational Foci 

I found that how a student interacted in class with the instructor, the mathematics, 
and his peers provided me with important data. When considered with material that 
emerged in individual counseling sessions, it helped me narrow and define that student's 
relational focus — in other words, his core relational conflictual pattern. By itself, 
classroom observation was certainly not adequate to identify students' entrenched 
mathematics relational patterns as they affected their mathematics mental health and 
prospects for success in PSYC/STAT 104. Observation seemed to even add to the 
confusion at times. However when I used the relational dimensions to organize my 
classroom observations, the data that were initially confusing often became important 
clues to students' core relational conflict (see Appendix E, Table E3). 
The Classroom and Issues of Mathematics Self 

During lecture discussions, students' ways of interacting gave some clues to their 
sense of mathematics self. Were they willing to reveal ignorance or only knowledge? 
Were they interested m growing, in performing, or in merely surviving? Lee and Robin 
were the most public, in different ways, in their attempts to grow and their willingness to 



189 

reveal ignorance for that purpose, indicating to me that they could have healthily 
developing mathematics selves and learning motivation for achievement. Others, like 
Autumn, Mitch, Karen, and Mulder, who only revealed their knowledge and were silent 
when they were uncertain, signaled a more fragile mathematics self. Autumn and Mitch 
acknowledged their performance motivation for achievement in their Mathematics Beliefs 
survey (as I expected from their other behaviors), while Karen, and Mulder surprisingly 
revealed more learning motivation. It seemed that their silence in class (except when 
certain) may have been self-protective, with survival taking precedence over their 
underlying desire to learn. Mulder, Floyd, and Brad all spoke and acted with confidence 
in their own knowledge that their exam grades belied. 

The student behavior 1 observed during problem-working sessions (particularly 
with respect to arithmetical and algebraic comfort and level of confidence tackling new 
material) gave clues to students' mathematical self development. In class discussions, 
Karen just seemed to want to survive. There were fiirther clues in problem- working 
sessions that her difficulties could be related to an underdeveloped mathematics self (e.g., 
her poor sense of decimals, poor operation sense, and low level understanding of the 
algebraic variable). 
The Classroom and Issues with Mathematics Internalized Presences 

A discrepancy between how the student was experiencing the classroom and the 
reality of the classroom was sometimes a clue to the effect of the student's internalized 
mathematics presences from the past skewing the present experience. Jamie serves as an 
effective example. Given the small class size, and the community-style seating 
arrangements, and the positive classroom emotional climate created by Ann, Jamie's 



190 

almost complete lack of participation (in fact, her quite successful hiding) was a clue that 
internalized past negative experiences might be skewing her perception of the present 
class and making her feel unsafe in a safe environment. Likewise Karen's observable 
defensive detachment and difficulty with the class in the beginning may have been 
related to her difficulty with separating herself from past experiences, especially that of 
previously taking the class. My awareness of the possible implications of these students' 
behaviors prompted me to explore further in counseling (see chapter 6). 
The Classroom and Mathematics Interpersonal Attachment Issues 

Students' attachment patterns to teacher and mathematics also became apparent in 
class. Kelly's behaviors indicated that she was anxious, disorganized, and dependently 
clinging to Ann and me; she seemed to have an insecure attachment to mathematics and 
to mathematics teachers. Although Lee seemed to have experienced a secure attachment 
to mathematics at times and had a history of generally secure attachments to mathematics 
teachers, she exhibited a lack of confidence in her ability to develop understanding on her 
own. This and her difficulty with Ann's approach showed in her spending up to two and a 
half hours a week with me in study groups and counseling while spending only about 20 
minutes doing homework on her own. Karen was detached and defensive and kept Aim 
(and me initially) at a distance — indicating the possibility of either a lack of secure 
attachments to mathematics teachers in her history or a traumatic severance of such an 
attachment with no subsequent reconciliation. Her confusion with decimals and her errors 
with simple arithmetical procedures in problem-working sessions indicated a lack of 
secure attachment to mathematics, almost certainly contributing to her expressed anxiety 
during these sessions. In contrast, Jamie's anxious and disorganized attachment pattern 



191 

seemed to imply prior positive mathematics experiences with intervening negative ones 
so that she was now uncertain and now had little sense of a secure base in the 
mathematics or in mathematics teachers. 

The ability to negotiate change was a particular issue for some students who were 
repeating the class, particularly Karen and Mitch. I used my knowledge of the statistics 
and my observations of Ann's teaching to build bridges from their past to their present 
experience and facilitate their adaptation to this new course. However, the main challenge 
was to help them acknowledge their conservative impulse reactions (cf Marris, 1 974), 
recognize the differences between the current and their previous class, and take 
responsibility for adjusting, rather than externalizing their discomfort by attributing 
responsibility to Ann. 

Whereas student's ways of relating with Ann and me seemed to fall into the 
category of attachment relationships, their ways of interacting with peers seemed to fall 
into the category of relationships of community (Weiss, 1991). These relationships of 
community were evidenced in how students related to each other — as social, 
independent, voluntary loner, or involuntary loner — and were apparent in problem- 
working sessions and in study groups. In mathematics counseling, I explored further how 
a student's pattern of relating with peers affected or was affected by his mathematics 
mental health. 

In the next chapter we will move from the classroom to the counseling setting 
where I describe and analyze in detail the course of mathematics counseling with three 
focal students. 



192 



' Excluding Ellen about whom I have no data. 

" I remained seated during these problem-working sessions in order to observe, but I also assisted students 
around me if they asked me. 

'" Pagano (1998) does provide his own links between concepts and procedures in the text, verbally and with 
diagrams, graphs, and illustrations (e.g., explaining the normal curve, pp. 81-86). He does not invite 
exploration nor pose open questions for his readers to find links themselves. 

" In the text, there is reference to statistical analysis computer software packages and optional companion 
manuals for MfNITAB or Statistical Packages for the Social Sciences (SPSS) that may be used with the 
text, but the only reference to them in the text is in chapter 1 (Pagano, 1998, p. 1 1 ) in a brief discussion on 
the use of computers in statistics. Pagano, in his preface to the 5* edition, notes that he had removed the 
cross references to computer software programs from the text at the request of teachers and students 
(Pagano, 1998, p. xix). 

The psychology department had developed a program of computer analysis projects, independent 
of a text, using an old version of MINITAB to be completed by PSYSC/STAT 104 students. Because of the 
accelerated timetable in the summer, instead of every student having to do each of 7 required MINITAB 
computer assignment modules, Ann required Module 1 for everyone, worth 2% of the final grade. The class 
then paired off to do one module per pair from modules 2 through 7 and these were presented at the second 
to last class. This was worth 8% of the final grade. 

" The text also addresses the issue of common misuses of and misconceptions about statistics in the seven 
"What is the truth?" inserts scattered through the text where the author links statistical concepts to an 
analysis of real-life mathematical or logical claims of advertisements, research reports, or news items in an 
attempt to link the text with and perhaps challenge the student's reality. However, the answer is given and 
there is no invitation for the student to examine his own beliefs and reactions to the material. Ann did not 
use these in class but instead distributed a copy of a newspaper article that she invited the class to 
critique — if there had been time. As it was she pointed out the errors in use of the statistics. 

" Lee challenged two of the four possible conclusions from a high correlation coefficient that Anne dictated 
fi-om the text (i.e., X caused Y and Y caused X) but eventually resolved this issue for herself by putting 
caused in quotes so as not to be associated with what she knew to be an erroneous step of attributing 
causation where "possible relationship" was the only valid conclusion. What Lee did not understand was 
that Ann and the text were correct in giving possible real coimections between variables that would lead to 
a high correlation, wiiereas Lee was rightly objecting to concluding that X caused Y or that Y caused X 
because of a high correlation between them. Ann did not have time to resolve this to Lee's satisfaction. 
Lee's subsequent response to a question about this showed that she had become conflised whereas on the 
pre test she had shown a correct understanding (question 16 on the post SRA, see Appendix C). In 
counseling 1 did not attoid to the real vulnerabilities of Lee's mathematics self-revealed by this situation, so 
we did not address her conflict in counseling. This situation may also point to her vulnerability to authority 
over reason. 

™ The worksheets, with the exception of the one factor and two factor x' worksheets, consisted of a 
question and, in some cases, an empty table v^dth column headings. However, the one factor and two factor 
X' worksheets were different. Ann had posed the question at the top of the page and then provided a step 
by step, fill in the blank, procedural format. Not surprisingly Lee objected strongly to it. Lee relied on 
working out the procedure for herself, in class, to guide her in the tests; she found the fill-in-the-blanks 
format confusing and distracting. She subsequently made an uncharacteristic and serious error on the two- 
factor X" on the test which she attributed to the worksheet. Instead of apportioning the expected 
fi-equencies proportionally among the four cells using the formula, /e= (Row Total)(Column Total)/(Total 
Observed Frequency), Lee apportioned them equally. Jamie and Robin both made an even more serious 
error adding the column and row totals to get a total expected frequency double the total observed 
frequency — clearly not reasonable if they had thought it through. These errors seemed to be not only 
related to the more directive worksheet that did not require the student to work his own way through the 



193 



procedure, but perhaps also to the absence of class or individual discussion of the mathematics or logic 
inherent in the formula. Ann stated that in other classes she had taught, students preferred the directive 
worksheets. 

™' Lee's relatively sound arithmetical and algebraic background seemed to help her in this, but there was 
not time in class for deeper conceptual explanations and connections. The lack of these made her anxious 
so that when she came to study group and individual sessions our focus was on connections and 
mathematical meanings (Level 4 on the Algebra Test, see Appendix H, Table HI and Appendix C). 

" Karen had arithmetical difficulties (for example, uncertainties about values of decimal fractions and 
placement of the decimal point) and her algebra background was shaky (Level 2 on the Algebra Test, see 
chapter 6, Tables 6. 1 and 6.2 and Appendix C). 

" The manipulation required is largely linear and usually direct, except, for example, when one has to find a 
particular score given its percentile rank in a normal deviate distribution, which is an inverse procedure 
requiring manipulation of linear terms. Ann didn't expect students to do this. In the least squares linear 
regression analysis section, which she did require students to do, they are required to derive a linear 
equation in two variables and use it to find particular points. This latter process requires only the 
substitution of numbers for variables in the derived equation. 

"" To transform the independent variable by translation and compressing or stretching in order to convert 
the probability density fiinction of the data in question into a standardized form whose area (i.e., 
probability) values are accessible on statistical tables in the text (or statistical software package). 

In statistical formulae the extensive use of subscripts as labels is complicated by the use of numbers, 
single literal symbols, and even variable expressions as subscripts (see discussion of Class 13). In addition 
in descriptive statistics the X(the mean of scores, ^Y, in a sample) in the first part of a course is a statistic, 
that is, a constant for that sample; the score JT is the independent random variable in this distribution of 
scores. However, in inferential statistics, the A' (sample mean) becomes the random variable because 
sampling distributions are distributions of the sample means of all possible samples from the population of 
a particular size (refer to the discussion on Class 13). Ann's sense of students' difficulty in understanding 
sampling distributions and her own may have stemmed, to some extent, from the lack of discussion in the 
text or elsewhere of this transition of the X from being a constant to being the variable (see From Exam #1 
through Class 12 discussion). 

'"'' I use social learner here to refer to a student's evidenced preference for collaborating with other students 
in contrast to preferring to work alone (a solitary learner or loner). This categorization should not be 
confused with Belenkey et al's connected versus separate knower which refers more to a student's 
preference for personal connection with the material being learned. It also should not be confiised with 
Skemp's (1987) categorization of relational versus instrumental mathematics learner which refers to a 
preference for conceptual understanding in contrast with a preference for procedural (only) competence. In 
addition to being a social learner Lee was also a connected and a conceptual (relational) learner, whereas 
Mulder, who was also a social learner, was a separate and procedural learner by preference or at least by 
socialization. 

Ann then addressed what she expected to be some anxiety about the class by telling of her own 
experience in learning statistics as a graduate student. In particular she referred to her metaphor of statistics 
as a beautifully painted mural with all the elements separate, distinguishable, and in their correct places in 
relation to each other. In an exam, under stress, however, it was as if the separate elements began to run 
together to form a horrible brown indistinguishable mess; she couldn't tell one procedure from another. 
She said she had recovered from this disaster, going on to master the subject at doctoral level. She went on 
to explain to the class how her experience of teaching this course a number of times had increased her 
confidence and her enjoyment of statistics and that she hoped the students would find taking the course an 



194 



"okay experience." She urged as a remedy to anxiety that students do their homework, study their notes, 
and ask for help until they had "over-learned" the material. 

" The only caution against this openness is the possibility of disturbing the trust of students in the received 
knowledge phase (the first phase of epistemological development) who believe that the teacher or text — the 
external authority — is the repository of all knowledge (Belenky, Clinchy, Goldberger, & Tarule, 1986; 
Perry, 1968). Such a belief tends to develop too in students enculturated in transmission, teacher-as- 
authority mathematics classrooms (see chapter 3). To help these students fmd their own ability to 
understand the teacher as she expresses uncertainty can simultaneously model exploring the mathematics 
and discovering it as a secure base. 

"^ I did call on Jamie in study groups and though that made her uncomfortable, in the small group she was 
able to respond. In a classroom setting when I am the instructor, my practice is to call on students for their 
responses in order to ensure that students who do not voluntarily participate are involved. With shy students 
who exhibit discomfort (and sometimes cognitive confiision) when called upon, I make prior arrangements, 
letting them know ahead of time of the question that I will be asking them to respond to. 

'™' Because this is an odd numbered chapter, 1 use "he," "his," and "him" as the generic third person 
singular pronouns. 

"^^ The feet that this extra exam was comprehensive, covering all the procedures and all the statistical 
concepts from the course ensured that it was nol equivalent to a course exam. Instead it was more difficult 
to do well on, especially without in-class review and specific preparation. Ann reported that no one in her 
prior classes had taken advantage of this offer so she was surprised that more than a third of this class (4 
students) chose to take it. My being available to provide preparation help was perhaps a factor. AH scored 
below their course average (at least one letter grade below). Two of the students scored just well enough on 
the exam for it to replace a lower course grade and ensure that they moved up into a higher final grade 
category (Lee from a B* to an A' and Pierre from a C* to an B" ). 

'"' My initial impressions of students were similarly affected. Meeting participants individually and 
gathering multimodal data about them modified my first impressions however. See also final discussion of 
student-instructor interactions. 

'" Starting at the left front and going anti-clockwise, Ellen and Pierre, Lee and Robin, Jamie and Catherine, 
Floyd and Brad, Autimm and Karen, and Mulder and I interviewed each other. As we were beginning, 
Kelly rushed into class late, so she joined Mulder and me. Kelly and I interviewed Mulder and Kelly 
introduced Mulder to the class, Mulder interviewed me and introduced me to the class. 1 interviewed Kelly 
and introduced her to the class. 

'™ The four types of measurement scale are: nominal, ordinal, interval, or rational scales. 

'"" The blank columns were labeled X - mean, and (X - meanf respectively This sheet was designed for 
students to compute the deviations of scores from the mean and then the squared deviations in order to 
compute the sum of squares (of differences of scores from the mean) and from that the variance and finally 
the standard deviation of scores from the mean. This procedure and 
thus the formula from it, namely/ IXX - mean)' (for a sample; for a population the denominator is A^ 

V n-l 
are labeled "empirical" because they reflect the actual process for finding how the scores vary from the 
mean. Ann alluded to the alternative "computational" formula that does not, and told students "I like this 
[empirical] way." 

™" Overall, an average score of 3.2 (on the 98 items of MARS) is at the 95* percentile (Suinn, 1972), 
indicating extreme mathematics anxiety (for further discussion of these scores, see chapter 6). 



195 



'^^ Each student prepared his own formula sheet to be used when doing part II of the exam, the 
"computational" part. We could include formulae and descriptions of symbols but not their definitions, as 
well as visual layouts for a procedure such as the labeled columns for finding the standard deviation, but no 
worked examples. 

"" 1 had obtained permission to use it from Dr. Garfield and from the Office of Sponsored Research's 
Institutional Review Board for the Protection of Human Subjects. See Appendix C. 

™" The cumulative frequency of the class group just below the class group from which you are trying to 
finding the score for the corresponding given percentile rank. To identify this correctly required students to 
first create a cumulative frequency column, interpret the subscript L to mean below (although L refers to 
lower limit of the current interval wtien it is the subscript in Xl), and then find the cumulative frequency 
immediately below the one for the interval in focus. 

'™™ To compare two independent sample means, the null hypothesis is that there is zero or no difference 
between the two population means (\x\ ^d H2 for independent samples), so that in the / test formula, (n, _ H2) 
= 0. For the population mean of the differences for correlated samples the null hypothesis states that there is 
zero mean of differences (hd)> or that there is no change, translates in the t test formula that hd= 0. 

'°'™' For Question 3 in Exam #3 the correct decision was to reject the null hypothesis and conclude that 
there is a relationship between amount of relaxarion and hot or cold baths. This decision is based on the 
magnitude of the / statistic being greater than the magnitude of the critical value of the t with which it is 
compared. 

'""'' Ann's thinking might have been, in this case, that the importance of coming to the correct conclusion in 
research justifies a severe penalty for the wrong one, in addition to the penalty already incurred for making 
mechanical errors. 

Robin asked "on task" questions and answered Ann's questions correctly approximately twice as often 
as she questioned or answered incorrectly, tentatively, or off task. This was not substantially different from 
Brad, for example, who made almost the opposite impression on Aim and me in class. Robin's almost 
constant frovm of puzzlement and flustered air seemed to be related to her relative difficulty with auditory 
processing of verbal material and her compensatory propensity to ask questions or check her understanding 
whenever she was uncertain that she "got" it. Robin also seemed to be exhibiting the well-dociunented 
tendency of women to be considerably more tentative about what they know than a man typically is (and 
Brad certainly was). 

'°™ In contrast are closed study groups to which students commit at the beginning and other students may 
not join following the commitment period. These groups then have a consistent membership. Lack of 
attendance may lead to a person being excluded. 



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CHAPTER VI 

UNCOVERING MATHEMATICAL RELATIONAL PATTERNS: THREE 
PSYCH/STAT 104 CASE STUDIES 

I have described the class in its context in the previous chapter; now is the time to 
zoom in on the courses of brief relational mathematics counseling with the participants 
from the PSYC/STAT 104, the focus of this study. What actually happened? As I looked 
at the participant and at me and at us in a way that was different, that is, relationally, and 
we explored the participant's relationality about mathematics as I supported her doing her 
statistics coursework, what did that look like? Was it different in process or outcomes 
from a traditional series of tutoring appointments? If so, how? In this chapter I present 
three counseling cases in order to address these questions. Initially I wrote each case as a 
profile of a student in the process of mathematics counseling within the context of the 
class. But then I realized that although the student is the focus of attention in traditional 
mathematics academic support, with this new relational approach I, as the counselor, also 
came into focus. It struck me that it was, in reality, we — the student and I, and our 
developing relationship — who were the object of this study. Before I present the cases 
though, I will briefly review the counseling activity in the study and explain fiirther my 
rationale for choosing Karen, Jamie, and Mulder from the ten. 

Each mathematics counseling participant and I undertook the task of 
understanding mathematics relational patterns (in particular central mathematics 
relational conflicts) and pinpointing issues salient to a good-enough resolution of that 
conflict while she' was taking the statistics for psychology course. The approach we used 
was different from the typical treatments in its focus on joint understanding: That is, 
students' class assessment results and survey responses became the object of discussion. 



197 

modification, and deeper mutual understanding rather than pronouncements that locked 
them in — in their minds and in mine. The relational counseling explored both conscious 
and unconscious forces the student and I were experiencing, and the cognitive counseling 
stressed continual conscious interventions using the insights we gained. 

The ten mathematics counseling participants had between three and eight 
individual sessions each, averaging close to five per person. I expected that only students 
who were anxious or saw themselves as "bad at math" would volunteer to meet with me 
for individual mathematics counseling. Instead almost the whole class signed up. The 
group included students who were extremely anxious, some who were not particularly 
anxious, and those who were somewhat ambivalent. Some wanted help with the 
mathematics while others who did not think they needed mathematics help signed up to 
help me with my research. Some might have accessed mathematics academic support if I 
had not been in the class; others defmitely would not. 

I found that the distinctions among the participants that were most indicative of 
the soundness of their mathematics mental health were the level of mathematics 
preparation (in terms of arithmetic [number and operation sense in particular], and in 
terms of understanding of the algebraic variable), which seemed to directly affect their 
mathematics self-esteem and interact with that to produce their particular condition of 
mathematics self It was in talking about their mathematics learning histories and seeing 
connections between those histories and their present patterns of mathematics 
relationship, that participants' central relational conflicts around mathematics became 
apparent. These conversations raised to the surface participants' and my awareness of 



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these conflicts and supported some resolution. They provided key factors both for the 
course and for their mathematics selves that could profit from brief therapy. 

THREE CASES STUDIES 

At the end of chapter 4 I alluded to my approach to choosing Karen, Jamie, and 
Mulder as focal participants. Here I will explain more fiilly. Karen and Mulder were 
mathematically underprepared students who acted quite differently but whose relational 
patterns seemed to stem from a similar source. They, Karen more than Mulder, were 
among the students most cognitively and relationally vulnerable to withdrawal, failure, or 
inadequate grades — the students whom mathematics learning specialists most struggle to 
imderstand and help in order to avoid disaster, often to no avail. Jamie, whose 
mathematics background was more substantial than Karen and Mulder's, was, however 
also surprisingly vuhierable to failure, even with relatively sound cognitive preparation. 
She had serious relational challenges that jeopardized her chances of success. Karen 
might, Jamie might not, and Mulder probably would not have accessed the traditional 
mathematics academic support offered by the college. Each had mathematics learning 
issues that emerged from different dimensions of their mathematics relationality. All 
three had learning styles that had affected their mathematics relational patterns differently 
and impacted how they were negotiating the present course. Though each is unique, taken 
together, they represented atypical range of student issues that the Leaning Assistance 
Center sees. 

I faced quite different challenges dealing with Karen, Jamie and Mulder and 
understanding myself in relation to them. I experienced Karen's holding me at arms 
length as a challenge but I also found it finastrating and worrisome — I had to be content 



199 

with her setting boundaries that I had to respect even when I beHeved they might be 
counterproductive to her progress. Jamie's shyness and obvious discomfort when in focus 
evoked my sympathy and protective impulses at the same time that I felt I needed to 
tiptoe around her, anxious that I might harm her. Mulder was opinionated and stubborn 
and he and I sparred — I found myself on the side of the opposition — which felt as if it 
included Ann, the instructor, and perhaps his Mom. Each taught me about myself as a 
tutor, a counselor, and a person; each learned about him or herself as mathematics 
learners; and we all overcame mathematical and personal challenges to achieve success in 
PSYC/STAT 104. Before I tell our stories I will quickly review the theoretical bases that 
formed the framework for the relational counseling I employed. 
Theoretical Bases and Case Presentations 

The theoretical bases for brief mathematics relational counseling were discussed 
in chapters 2 and 3. Essentially my approach involved embedding cognitive 
constructivist, problem-solving, strategic tutoring in a brief relational conflict counseling 
framework. This was a dynamic process that differed considerably from participant to 
participant. What emerged from each participant's course of counseling, however, was a 
common phenomenon that, while providing me with a pivotal key to understanding his or 
her central relational conflict, also gave me a central organizer for presentation of these 
three focal cases. That key was each participant's metaphor for mathematics or 
themselves doing mathematics. 

In presenting the cases then, after introducing the participant and me and our 
relationship, I begin with the participant's metaphor and discuss the mathematics 
relational implications of the metaphor that we discovered. This discussion leads into 



200 

consideration of the participant's mathematics relationality and how we understood and 
worked with it through the course of counseling. The participant's present ways of 
relating with me, the instructor, and mathematics — his or her relational patterns — 
illuminated each of the dimensions of relationality that Mitchell (1988, 2000) identified 
and that I adapted to the college mathematics learning support context, namely, the 
mathematics self, mathematics internalized presences — teacher/s (or parent) and 
mathematics, and teacher and mathematics attachments. Disturbance in the development 
of one or more of these dimensions led, for each, to present mathematics-related 
emotional conditions, understanding which, in turn further clarified for us the 
participant's relationality and central relational conflict. Understanding a participant's 
central relational conflict, in the context of his or her mathematics relationality, helped 
me develop a counseling focus. Finally, I follow discussion of this counselmg focus with 
a summary of the course of counseling, session by session, to illustrate the processes, 
demonstrate the changes we made, and present outcomes. 

KAREN'S COURSE OF COUNSELING 
Karen ''had to pass [PSYC/STAT 104] this time." I found this out by the vending 
machines during break of the third class meeting. As we were choosing our snacks, I 
commented on her being one of only two in the class apart from the study group to have 
done an extra assigned homework problem. She told me then that she needed the class for 
her psychology major but had failed it two summers previously. She sounded somewhat 
desperate. Even then, before I had met with her one-on-one, after observing her only over 
two and a half class meetings and despite her doing the homework problem, 1 had an 
ominous feeling about how she would do. 1 had already observed her keeping all 



201 

classroom personnel at arm's length, including the instructor and me. She seemed to be 
positioning herself defensively. Karen's working alone during in-class problem-working 
sessions seemed intentional and she had not attended the study group; instead she came 
early to the classroom, sat at the back, and worked on her own while the study group 
worked with me on the board (see chapter 5, Figure 5.1 and Appendix F)." 

I was not sure how to interpret her signing up for counseling. It seemed 
incongruous with her distancing stance but consistent with her expressed need to pass this 
time although she did limit herself to signing up for once every other week not once a 
week, which was the option I expected from someone who had already failed the class. I 
wondered how it would be. I wondered if mathematics counseling would be any use. I 
was worried that the task, that Karen's needs and her defensiveness, would overwhelm 
both of us. I was anxious that Karen would especially resist my relational counseling 
approaches but knew that these approaches had the potential to help her succeed this 
time. 

What I did in counseling was to go ahead anyway, tackling the statistics and 
working side by side with her as we looked at the mathematics, I heard her voice and 
together we challenged her negative sense of herself doing mathematics. At the same 
time we evaluated the grounds for her defensive relational patterns. And I realized that 
my initially overwhelming negative sense of her doing mathematics was also challenged. 
Karen made better and better choices as she discovered a competency she had not 
previously recognized and teacher support she had initially rebuffed. Her expertise and 
confidence increased and her grades improved from a D' on the first exam to B^s and 
'A's at the end with an overall 'B' for the course. 



202 

As I worked with Karen I learned to attend to and manage my countertransference 
reactions to Karen's initial defensive negativity. I experienced her transference as her 
teacher who would "know" as she did the severe limits on the mathematics she could do. 
She seemed very negative about her prospects for learning mathematics. "That's how I 
am. I can't/won't be able to... I can plug in the numbers but I don't know why ..." 
(Sessions 1, 2); and I felt firmly rebuffed as I imagined her former teachers did if they 
tried to make a difference. In my countertransference I surmised that Karen's teachers 
before me may have accepted as I had begun to do that she was unlikely to succeed; this 
made me feel desperate and overwhelmed. But I (and she) challenged my 
countertransference reaction and I chose to believe and act differently. By looking at 
Karen from a relational perspective, I was able to help her find a real but underdeveloped 
mathematics self and develop it further. By the end of the course, neither of us thought of 
her any longer as someone who could not do mathematics. I was also able to challenge 
her defensive detachment from Ann and me; Karen began to experience us as secure 
bases on whom she could rely and from whom she could eventually venture out on her 
own. Indeed, Karen still had mathematical challenges, true, but she could face them 
knowing that she had found herself able to do well enough to succeed in this course. 

Karen was a tall, blond, 22 year-old white, elementary school assistant teacher 
who had dropped out of State University after a year and a half and was pursuing her 
degree part-time at Brookwood State. She was the first in her family to pursue a 
bachelors' degree although she reported that her parents had taken some post-secondary 
technical courses. Karen wanted to become an elementary teacher and was majoring in 
psychology but only because the university required prospective elementary education 



203 

students to major in a non-education field (Class 1 ). In beginning this second attempt to 
pass PSYC/STAT 104, Karen stated that she hoped for a B but expected a C/B (Pre-Test 
Mathematics History Survey). 

Karen's Metaphor: Mathematics as Cloudy 

As the most representative of Karen's metaphors for mathematics: "black," 
"stormy," "cloudy," "bear," she chose "cloudy" "because there are some aspects of math 
that are more clear to me, but mostly math is my worst subject and has always been hard 
for me to understand" (College Learning Metaphor Survey). That she chose what seemed 
to me to be the mildest image from her list surprised me. Karen's rather diffuse, 
somewhat depressed, image of a cloud seemed to contrast with her almost aggressive 
defensiveness, which made me expect her to select the image of defending herself against 
a bear rather than seeing her way through a cloud. Still "cloudy" did seem congruent 
with what I sensed as a resigned desperation, which to me felt as though she was 
experiencing groping around in a cloud as fruitless. 

As we proceeded with mathematics counseling I understood better what Karen 
meant by her distinction between "more clear" and "cloudy" mathematics — it was partly 
about the type of mathematics: "I'm better algebraically than I am geometrically. . .1 can't 
do geometry at aU"(Session 1). But perhaps it was even more about Karen's sense of her 
own limits: "[Mathematics is] my worst subject ...always hard for me to understand," 
She clarified this fiulher by responding "nothing" when I asked what she understood 
about a new concept, explaining "see that's how I am"(Session 2). Her use of the word 
"always" seemed to mdicate a long-term and global negativity about herself as a 



204 

mathematics learner. When I asked her about it, Karen confirmed that 'always' meant 
"Back — all through school" even in first grade. 

Karen's Mathematics Relationality 
Student-Mathematics Relationships: Karen 's Mathematics Self and Cloudy Mathematics 

Since first grade Karen said she had found mathematics cloudy, "hard to 
understand." In mathematics counseling when I asked her as an adult about her 
mathematics metaphor her first statement to me was, "I hate math." I wondered what her 
experience of mathematics had been through school for this to be the outcome. 

JK: How have you been historically with math, you know, through the grades? 

Karen: It depends on what kind of math it was. If it was like geometry or 

something like that, I did horribly {okay} but Algebra and Algebra II, I 
didn't do too bad on. I just don't like math {yeah} at all. I never ever, ever 
have. 

JK: Even in elementary school? 

Karen: Nope I've always, I like reading and writing not math or science 

At our first meeting Karen expressed an antipathy to mathematics requiring 
interpretation of visual material (e.g., graphs and diagrams): "I'm better algebraically 
than I am geometrically. I can't do geometry at all" and later "I hate those bell curve 
things." Karen told me she had turned away when Arm had drawn a bell curve in the last 
class (Class 3) because she disliked them so much. She believed, however, that 
conceptual learning of algebra was beyond her. "If it's algebra, and it's just a matter of 
plugging numbers into certain formulas, I can do pretty well with that... I can plug all 
those things into that and I have no idea why, or what that means" (Session 1). I 
mentioned to her that the study group had been working at understanding how and why 
the percentile point and rank formulas worked and suggested that she might feel more in 
control if she understood. Karen demurred, "Not necessarily; sometimes it's easier if I 



205 

don't know why — I can just do it" (Session 1). I interpreted Karen to be saying that an 
attempt to understand the procedure might undermine her tenuous grasp of how to do it. I 
wanted to help her discover that she could understand, at least how this formula made 
sense, but she did not want to risk it. 

It was clear to me from these data that Karen's mathematics self-esteem was quite 
low. She communicated that by describing her low confidence in her mathematics 
capabilities ("That's how I am."), her low expectations ("I'll bomb the conceptual 
portion."), and ahnost global negativity — possibly to protect her mathematics self from 
further disappointment. She had little of what self psychologist, Kohut (1977) calls "a 
storehouse of self confidence and basic [mathematics] self-esteem that sustains a person 
throughout life [in the mathematics classroom]" (p. 188, footnote 8). 

How realistic or accurate were her negative self judgments? Did she have enough 
arithmetical and algebraic competence to build new learning on? Was she actually more 
firmly attached to mathematics than she believed or felt? I gathered a more systematic 
picture of Karen's arithmetic and algebra competence during posttesting and this 
confirmed what I had found through observation of Karen's work in counseling and the 
classroom during the course. Particularly with fractions and decimals, Karen's number 
sense and operation sense were very weak (see Table 6.1). This made it difficult for her 
to evaluate the appropriateness of the numerical results of her data analysis or to 
troubleshoot her work in order to self-correct an error. In addition, Karen was operating 
at a level 2 understanding of the algebraic variable, and here she was the lowest in the 
class (see Table 6.2). That meant that she was able to coordinate operations with letter 
symbols as objects but that she did not understand letter symbols as specific unknowns or 



206 

generalized numbers (and in some cases as variables) and could not coordinate two 

operations on them.'" Given this Karen was likely to find understanding and using letter 

symbols in complex statistics formulae difficult. How she prepared the formula sheets to 

use for exams could be crucial. 

Student-Teacher Relationships and Cloudy Mathematics: 
Karen 's Mathematics Struggles 

I wondered how Karen's mathematics self development had proceeded for her for 

her to have such crucial mathematics deficits and to feel so negative. What part had her 

family and teachers played? Perhaps there was a family connection to her "always" 

fmding mathematics "hard... to understand", I thought. It seemed that she had never 

reflected on it before, but now she began to see it. 

JK: What about your parents? Are they more like that [reading and writing, 

not math people] too? 
Karen: Yeeeah? [considering] Yeah=, yept (Session 2, see chapter 4, Figure 4.3 

for coding conventions used) 

Because of what I experienced as Karen's reticence in talking about anything 
personal, I took the enthusiastic agreement I heard in "yept" to indicate that yes, she had 
experienced her family culture as one where her not having an interest in nor doing well 
in mathematics were accepted, perhaps even expected. I brought it up later and Karen 
said, "I'd say we're more of the reading, writing type, the whole family" (Session 5), thus 
confirming her sense that doing well in mathematics was not part of her family scene. 
When she did do well on an exam (Exam #3) they were all surprised and delighted at her 
success. 

What about her teachers, then? What was their part in the development of Karen's 
mathematics-as-cloudy self? I asked her: 



207 

JK: Any teachers in math who, you know, who made you feel bad or better 
about yourself? 

Karen: No, not really. I mean 1 was never like worst in the class, you know. I was 
always in the middle, middle to lower scale but I suppose I concentrated 
less because I didn't like it as much so, you know? (Session 2) 

Karen seemed to have managed to get by in class by being unremarkable. She was 
not the worst so she did not attract negative teacher or peer attention, and she was 
certainly not the best. But it seemed that she had not received positive attention either. 
She had managed well enough to avoid attention, despite her perhaps defensive 
"concentrate [ing] less." If how she was relating to Aim and me was any indication, she 
had defensively kept her distance from them and teachers had let her be, accepting her 
limitations as real and essentially neglecting her mathematics self development. This in 
turn likely led to Karen's blaming herself, seeing herself as intrinsically bad (at 
mathematics) and not seeing the teacher as responsible (cf Fairbairn, 1 972). 

Confirming this, when I inquired whether there had ever been a negative incident 
with a teacher she shifted the answer to herself by implying again that her present 
mathematics situation was of her own making: "I was not interested in math at all. I don't 
like it. That's why I don't do as well" (Session 5). Karen seemed to be using lack of 
interest to avoid acknowledging what she really believed to be the reason: her underlying 
lack of ability. Karen never spoke of a relationship with a mathematics teacher in either a 
positive or negative sense. The only teacher Karen spoke of at all was her instructor from 
the first time she took PSYC/STAT 104, and then it was to compare her teaching 
approach with Ann's. 

I considered the absence of direct information from Karen about her experience 
with teachers, despite my probing, and realized that how she related to Ann and me in the 



208 

present course might give me the clearest sense of the relationships she had with 
mathematics teachers through the years. Karen was upset that Ann's teaching was unlike 
that of her previous teacher for this course whom she described as "more thorough." She 
believed that Ann would, nevertheless, expect her to know and use all the material in the 
text even if it had not been covered in class. Later I realized that the discrepancy Karen 
found most disturbing between the teachers was that her previous teacher had 
demonstrated on the board how to do each type of problem (perhaps her idea of 
"thorough") while Arm had each student tackle the problems herself in problem- working 
sessions that were sometimes lecture-guided but more often accomplished with her 
roving coaching help. Because of Karen's lack of confidence in her own ability — based 
on her low mathematics self-esteem Ann's approach made her feel anxious and 
insecure despite what I perceived to be Ann's adequate coaching support. Most 
prominent for Karen seemed to be a sense of Ann's not being there for her in a way she 
felt she needed. She seemed to feel abandoned. Past experience with mathematics 
teachers appeared to have promoted her adoption of defensive detached patterns that 
seemed to have been activated in this class. 

At our July 10 interview (Interview 2), Ann had expressed disappointment with 
her relationship with Karen: "I thought we would be closer." Karen sat as far at the back 
of the classroom as possible and she did not connect with Ann outside of class time. On 
several occasions (at least once in Ann's hearing) she expressed hostility about her 
perception of what had been said about what to expect on the next test in contrast with 
what Karen believed should have been said. This had to contribute to Ann's sense of 
Karen's hostility and deliberate distancmg. I considered this aggressive detachment to be 



209 

largely unconscious rather than deliberate on Karen's part. Keeping her distance seemed 
to be Karen's established defensive way of negotiating a situation that exposed her 
vulnerable, underdeveloped mathematics self 

I began to believe that Karen had not ever developed a secure attachment to a 
mathematics teacher. No mathematics teacher had offered herself as a secure base in a 
way that she felt safe to connect with. She had learned to care defensively for herself and 
expected little from the teacher. Such low expectations seemed to have made her angry 
and anxious, even hopeless, because she knew she did not have what was necessary to do 
it on her own and she needed support from the teacher. Aim's and my experience of her 
defensively holding us at arm's length suggested that her demeanor may then have 
become a factor inhibiting even good-intentioned teachers from reaching out to her. 
Karen's defensive distancing may have been exacerbated in the college setting by the fact 
the she was the first in her family to go to college. It was unfamiliar territory and she did 
not have family experience and advice to help her negotiate it. 

Emotional Conditions: Anxiety, Learned Helplessness, or Depression? 

How did Karen respond emotionally to what seemed to be the underdevelopment 
of her mathematics self? Was her reaction consistent with a diagnosis of 
underdevelopment of mathematics self, expressed in underconfidence and defensive 
detached relationality? Were her emotional responses interfering with her approach to 
PSYC/STAT 104 to an extent that warranted emergency attention? The way I 
experienced Karen at the first session felt confrising — I experienced her anxiety, 
negativity (even hopelessness) and anger. 



210 

Anxiety 

Karen admitted to being very anxious before the first test. Her scores on the 
Mathematics Feelings pretest survey confirmed that she consciously experienced 
excessive anxiety in mathematics performance and testing situations. On her Survey 
Profile Summary I had circled all three anxiety scales because they were all at almost the 
top of the class range (see Appendix K, Figure Kl). The combination of her Abstraction 
and Number anxieties, however, especially in conjunction with what I had observed of 
her issues and approach did seem to be directly related to her underdeveloped 
mathematics self, particularly her inadequate number and operation sense and low level 
understanding of the algebraic variable (see Figures 6.1 and 6.2). Her testing anxiety 
(second highest in the class) seemed also to be related to the inadequacy she felt when 
she tried to recall how to do procedures she dimly understood. At least for the first exam 
Karen's inadequate practice and unstrategic preparation contributed considerably to her 
heightened anxiety. It seemed that the anxieties Karen experienced in mathematics 
situations were normal reactions to threatening situations for which she felt inadequate. It 
also seemed that her anxieties could be considerably alleviated by more strategic and 
thorough preparation. 
Depression and Helplessness 

Karen expressed negativity about her mathematics self, mathematics, and this 
class. I analyzed her responses to the Beliefs Survey that she completed during the second 
class for underlying beliefs or constellations of beliefs that could better pinpoint her 
negativity as well as others that could show healthy positive orientations. Karen's 
average pre-score on the learned helplessness versus mastery orientation scale was 



211 

worrisome: below the middle of the scale it was the third lowest m the class. On all three 
belief scales, her responses fell below the class average. Nine of Karen's 14 Learned 
Helplessness vs. Mastery Orientation responses were 2 or below, reflecting her belief that 
learning mathematics involved having to be taught and then memorizing different 
procedures for each new type of problem. This belief would make her helpless if she did 
not memorize the right things. On the mastery oriented side, although she agreed that 
some people could do mathematics while others could not, Karen believed that her 
mathematics ability could improve, so it seemed that she was not locked into a frxed trait 
belief about this ability. Karen also reported that when she could not immediately do a 
problem she would not assume she could not do it and give up on it, and she usually tried 
to understand the reasoning behind mathematics rules. Karen's negativity about her 
mathematics self, world, and future did not preclude an underlying hope in the possibility 
of change; she also had a view of herself not giving up when learning was difficult (see 
Appendix K, Figure K2). 

Karen's responses indicated that she was more motivated towards learning than 
performance.'^ This surprised and encouraged me for Karen. Her focus was not just on 
results; she wanted to understand the material. She did believe mathematics to be more 
procedural than conceptual but her beliefs were not extreme (just below the midpoint) 
and with her expressed learning motivation and strategic support to find she could make 
the conceptual connections it seemed possible that her beliefs would improve (see 
Appendix H, Table H3). 

Karen's responses over time on the JMK Mathematics Affect Scales, however, 
lent further weight to a diagnosis of entrenched negativity even depression. To monitor 



212 

her negativity/positivity Karen filled in the scales at every counseling session except the 
first. Karen's responses at the end of the second session were negative, all seven 
responses falling at or below the mid-point. She was very much discouraged about her 
problems with mathematics and she would withdraw from the current course if she could. 
She expressed moderate to severe negativity about her mathematics self (scales 1, 2, 6, 
and 7), about her current mathematics world — the class (scales 1 , 2, and 4), and about her 
mathematics fiiture (scales 3 and 4) the three spheres Beck (1977) found to be significant 
for people suffering from depression. As the course of counseling proceeded and Karen's 
responses on the JMK Scales did not improve in proportion to her improving grades, my 
awareness grew that it was mathematics situational depression (and related learned 
helplessness) rather than anxiety that Karen was struggling with (see Figure K3, Table 
K3, and Appendix B). 

Identifying Karen 's Central Relational Conflict 
As we began Session 1, 1 was already drawn into Karen's anger and anxiety. I 
wanted simultaneously to rescue her from her plight and to defend Ann, the obvious 
target of her anger. So that she might not be angry with me too, I tried to be on her side, 
the fair, reasonable teacher she believed Ann wasn't. She kept her emotional distance 
from me too though as if I were on the side of the opposition. I did not want to believe 
her view that she was incapable of becoming more than a procedural mathematics 
learner, although I worried that the time-limited situation might force me to help her 
succeed only procedurally, thereby making her feel as if I agreed that she lacked the 
conceptual ability. Her view of herself as a mathematics learner seemed to be globally, 
diffusely negative, as if her mathematics self barely existed. Though she was trying to 



213 

contain it using external means (blaming Ann, formula sheet, last minute tutoring), 

Karen's sense seemed to be that this exam and this course were out of her control since 

there was little inside her to draw on. 

Karen's responses on the JMK Mathematics Affect Scales, taken with her low 

indices on the Learned Helpless/Mastery Oriented Beliefs scale, her "cloudy" metaphor, 

and her defensive detached stance in relation to peers, Ann, and me, pointed to a 

diagnosis of moderate empty mathematics depression (cf Kohut, 1977, and see chapter 3, 

pp. 91 ff.). This likely stemmed from Karen's deep sense of an underdeveloped 

mathematics self rooted in her poor mathematics preparation and low self-esteem. Her 

central relational conflict seemed to be between her strong desire and even need, to 

succeed in this course and her fear that there were powerful forces outside her control, 

including her own inadequacy and the instructor, which conspired to thwart that desire. 

Her significantly underdeveloped mathematics self seemed to be the chief conspirator. 

She seemed to be projecting her fear of her own inadequacy onto those around her. 

Karen and Me: Dealing with the Clouds Now: 
Relational Counseling for Karen 

The Focus of Relational Counseling 

I realized that, relationally, I had to provide myself as a guiding hand for Karen 

to safely negotiate her way out of the clouds that she had felt trapped in. To help Karen 

resolve her conflict I had to offer good-enough mathematics teacher-parenting to support 

the emergence and development of a firmer mathematics self that could succeed in the 

class. I planned to challenge her all-or-nothing thinking by mirroring her sound thinking 

and achievements and at the same time I would provide myself as a mathematics parent 

image that she could idealize and realistically incorporate into her increasingly competent 



214 

mathematics self almost like her internal mathematics guide. 1 expected that this 
development should go some way towards alleviating Karen's empty depression and 
underconfidence. 

I would have to work at overcoming Karen's emotional distancing enough that 
she would accept my mirroring, though. To do this I had to resist agreeing with her about 
her mathematics hopelessness. Although her transference of past teacher relationships led 
me to believe that her low confidence was realistic, I had to resist that interpretation and 
instead see it as unrealistic underconfidence; Karen was capable of doing mathematics. It 
seemed crucial that Karen become free to avoid repeating her past experiences of doing 
poorly in mathematics classes and failing PSYC/STAT 104. Importantly, this would 
involve helping her recognize and take advantage of Aim as a secure mathematics teacher 
base, rather than a neglectful but demanding teacher from the past. 

Although her angry anxiety was a potential focus I decided that it was a symptom 
rather than the root of her difficulties and could be ameliorated by helping Karen deal 
more effectively with her sense that external forces controlled her course outcome. I 
hoped that as her sense of her own competence grew, she would be increasingly able to 
take more responsibility for strategic exam preparation, she could seek help from Ann or 
me in a more timely manner, and she could make more effort to understand the 
mathematics conceptually. 
The Focus of Mathematics Tutoring 

Mathematically I would provide myself to Karen as a mathematical co-explorer 
with a flashlight and other tools that we could use to find our way through the cloudy 
terrain. Given Karen's multiple mathematical concerns and her evident course 



215 

management difficulties, I found that identifying a strategic mathematical focus initially 
overwhelming but I soon focused on Karen's underdeveloped mathematics self I decided 
to mirror back to her what I saw as her strengths in mathematics and her positive 
approaches to the course. This was likely to help her begin to see her mathematics self 
differently. I also set out to nurture and coach that developing mathematics self not only 
helping her to develop further mathematics understandings and competencies but also to 
recognize herself developing them. Then she might see herself finding her way through 
the clouds into the clear light to day. 

It seemed that if I worked beside Karen as she mastered new procedures 
introduced in class and helped her link them to the concepts, and if she practiced she 
would be able to understand enough and do new problems; she needed to also recognize 
that she could. Karen's motivation for this deeper work could come from seeing her 
growing ability to grasp these links herself Karen also needed to develop strategic 
structures (guide ropes to hold onto in the clouds) to compensate for her underdeveloped 
algebra, number, and operation sense. 

Recognizing the useflilness of connecting the conceptual portion with the 
computational part would give her increased control. The primary focus needed to 
include her developing skill with letter symbols. Karen's mathematics self was affected 
by her poor facility with decimals and percents as well as her underdeveloped number 
and operation sense. Tackling this would, of course, depend on the time and emotional 
energy Karen was willing to invest. 



216 

Karen's Course of Counseling: Session by Session 
(see Appendix K, Table Kl for Karen's schedule) 

Karen 's Sessionl 

Karen came before her first appointment to drop-in and I observed her desperately 
trying to practice problems she had not yet gone over. It was at drop-in that she angrily 
denounced Ann for expecting the class to know all of the material in chapters 1 through 5 
even though she had not covered it all in class. Karen had not gone to work that day 
because she was not feeling well and it seemed that she had spent some time scanning the 
chapters for material for her formula sheet (cheat sheet, she called it) and had become 
increasingly upset as she found unfamiliar formulae and concepts. 

Karen felt that Ann had not been clear about what would be on the exam so 
studying the right material felt beyond her control. Karen did not interpret Ann's Exam 
#1 study guide, her presentation of all the material in her notes, and having the class work 
through specific problems from each chapter as likely cues that this was the material that 
would be tested. All of the students were feeling some anxiety about this first exam, but 
Karen seemed to be particularly misreading the situation. I wondered whether her failing 
experience in the previous PSYC/STAT 104 class was so prominent at this point that it 
was interfering with her ability to read the cues Ann was giving. 

Her appointment with me was at 4:00 p.m. at Greenville campus and the exam 
was scheduled for 6:00 p.m. at Riverside campus. I had her continue with the problem 
she began in drop-in. I had reassured her that, based on the exam review guide Ann had 
distributed, and my sense that Arm had been careflil to cover in class all that she would 
examine, that her she could safely ignore the other material in the chapters and erase that 
unfamiliar formulae from her sheet. She was already confident in the direct process of 



217 

finding the percentile rank of a given score procedure from chapter 3 of the text. This was 
the one she had done correctly for homework: "Percentile rank, I've never gotten one of 
those wrong." But she was not confident of this inverse percentile point procedure, the 
one that we had done in class. After Karen did another of these to reassure herself, we 
checked the exam study guide for the list of symbols. We reviewed her understanding of 
the symbols to be tested for both name and meaning and she was quite confused." Karen 
was not aware of the Greek versus "English" (Roman) letters distinction between 
population parameters" and sample statistics,"" which I showed her. Although pleased 
with this organizing idea for symbols and formulae, Karen was still somewhat 
overwhelmed with the discussion of the concepts the symbols and formulae represented. 

As we proceeded I began to understand that not only was algebra cloudy to her, 
arithmetic was too. In finding 50% of 54"" Karen was content with 1 .08 (she had divided 
54 by 50) as an answer; it became clear she did not understand percents, not even a 
benchmark generally known.'" I wondered how pervasive were her arithmetic 
uncertainties and what effect that might have on the current course. With less than half an 
hour to go Karen announced "I have no clue on chapter 4 or 5." I accepted that global 
statement on face value and anxiously joined in her desperate but seemingly impossible 
race against time to cover that material before the exam. 

Karen 's Session 2 

"Horrible!" was Karen's response to her 62% on Exam #1. She was disgusted 
with getting the range wrong when she knew it; it was such a "simple concept!" Karen 
also reminded me that she hadn't been feeling well. The next exam was to be in two days 
time so I felt some urgency to begin breaking down some of Karen's negativity towards 



218 



Ann so she could take advantage of the structure and support she was offering. I also 

planned to help Karen recognize what she could do to begin to break into her global 

negativity about the rigid limits she placed on her mathematics self 

Right away I asked her: 

JK: How did you react to the exam itself? Better than you thought it was? 

{Yes} just what she [Ann] covered {right} rather than the whole book? 
K: Rightt 

Karen agreed without hesitation that her fears before the first exam [that Ann 

would examine material not covered in class] were baseless but later in the session she 

brought up an assigned problem she had struggled with at home that she was pretty sure 

Ann had not covered in class. 

Karen: It's number 13 {number 13} right, and I don't remember doing that, using 

this formula. 
JK: Is this? Is this? No [hesitating] 
Karen: So we don't even need to do that one then? 
JK: Where is this? Is this on the list? Is this on our list to do? Do we have that 

on our list? {Yeah} you can do it but [the problem m'os expecting student 

to go beyond procedures taught in class] 
JK: Where's your list? I saw you had it before. You seem to be neat, keep your 

things in order. 
Karen: Chapter 6 one through i *. No, it's nooot (-). See, I don't pay attention (-) 

{laugh} 
JK: She's fairly careful which ones she picks [to assign as homework 

problems]. So I was thinking, "Why would she give us that?" So number 

14, let's do number 14 (Session 2) 

Karen had further good evidence that her fear that Aim might set her up with 
impossible tasks were ungrounded and I was able to take advantage of the situation to 
help Karen notice Aim's thoughtflil planning designed to avoid such student frustration. 
Maybe Ann (and mathematics teachers) was more trustworthy than she thought. Maybe 
Karen could begin to consider trusting her. 



219 

Exam #1 Analysis 

When we analyzed Karen's exam together she had done better than she had 
expected on conceptual section of the test on that section. She only missed one out of the 
8 symbol questions" and contrary to her expectation "If anything, I'll bomb the 
conceptual", was correct on 75% of the conceptual questions — it was the computational 
section she failed. 

Questions involving decimals gave her trouble,"' so I advised Karen to arrange an 
extra meeting to do decimal exploration, "" since understanding and computing statistics 
involves a lot of work with decimals, it was likely that her anxiety and negativity were 
linked to her arithmetical uncertainties, and we were fmding that Ann stressed 
arithmetical accuracy in her grading. 

Karen had calculated all but one of the initial basic procedures accurately, and 
also succeeded on the direct, percentile rank procedure that she had practiced thoroughly. 
She was one of only four in the class to get this question entirely correct. Her response to 
the inverse find-the-percentile-point procedure showed her understanding of the concept 
but her anxious practice at drop-in and in the counseling session was not sufficient for her 
to reproduce the required procedure and she earned no points. She saw that what she 
failed was material she had not practiced at all from chapters 4 and 5. 

This analysis revealed a mixed picture. Karen saw evidence that when she 
practiced sufficiently she could succeed and her global negativity seemed unjustified. 
Karen had not come to a help session early enough for Exam #1 but this session was two 
days before Exam #2. Karen had realized she had not focused strategically for the first 
exam so we planned to focus on problems like the ones done in class. Her defeatism 



220 

about her ability to do the mathematics and interference caused by repeating the class 
seemed to have contributed to her difficulties on Exam #1 so in this session I began to 
address these issues. 

I asked Karen how her approach to the course work was different from what she 
did for course she had failed. She seemed taken aback by my question. Her response was 
"I think I just like, it just took me a while to get back into the [course] you know?" 
indicating that she realized that she had begun preparation for Exam #1 later than was 
wise. This seemed to contradict an earlier claim that she thought that she had been 
prepared for Exam #1. Now it sounded as if she might be revising her sense of what 
adequate preparation for an exam should entail for her. When I asked about her grades 
the first time she took PS YC/STAT 1 04, 1 discovered that Karen had all her course 
materials with her, including her test scores. The current test score 62% was considerably 
higher than her 47% on the first test then, she had succeeded in getting one of the most 
challenging questions correct, and she had overcome her confusion about the symbols, all 
of which began to break through my internalized negativity about Karen's chances. 

To prepare for Exam #2 we looked at the material that would be tested. When I 
asked her what Pearson's r was, she responded, "Nothing (-). . .See! That's how I am. I 
just plug in the numbers . . . That's why I have so much trouble." Her pronouncement 
indicated a significant change from her earUer defensive response that it might be better 
not to understand why. Now she conceded that not knowing was causing her difficulties. 

Now I wanted to help Karen see that she could make her way through the clouds 
and see clearly for herself I used a modified cognitive constructivist tutoring approach 
and kept alert for relational opportunities to mirror her competencies. I provided myself 



221 

as a model (a parent). The parallel modeling approach that we used looked much like 
best-practice traditional tutoring but had the added effectiveness of intentional relational 
attention. 

As we individually set up and solved the problem side by side I talked through 
it."'" We began by constructing a scatter plot of the data'"" and focused on identifying the 
independent and dependent variables. I waited for Karen's decisions before revealing 
mine. In the process, Karen found that her new understanding contrasted with her prior 
confusion in class when Ann had briefly demonstrated the scatter plot construction 
process, "Some of the points, either she [Ann] didn't do it right or I don't know where 
she got them from." Although she was implying that not understanding it in class could 
have been Ann's "fault" she also seemed to be conceding that it could also have been her 
own issue. 

Karen graphed the coordinate points without difficulty, but she was in trouble 
once the scatter plots were drawn. As I questioned, coached, provided prompts, and 
worked the problem beside her, Karen explored the relationships among the symbols and 
their graphical representations and meanings. She gave no hint of her earlier anti-visual 
position. She even reluctantly revived her hazy knowledge of coordinate graphing of a 
straight line and explored that further, both graphically and algebraically. We did not 
have enough time to calculate a standard error of estimate but Karen seemed to feel less 
anxious about the upcoming exam. She had a much better idea of what to expect, she had 
understood material she did not think herself capable of, she had two more days to 
prepare, and she was a little more assured of Ann's care and good intentions. But just 
before we left, Karen filled in JMK Mathematics Affect Scales (see responses labeled 2 in 



222 

Figure K3) and her responses were very negative (see discussion of Depression or 
learned helplessness above). 

Karen 's Session 3 

Before Karen's Session 3 I had interviewed Ann and Karen arrived just as she 
was leaving. Ann asked her how she was doing with her MINITAB computer module and 
Karen had some questions so Ann offered to go with her to the computer lab to resolve 
them. When they returned there was only half an hour left for our session. Ann resolved 
Karen's concerns about materials for her presentation and left after she told us of the 
research project she was launching the next day using an audiovisual presentation to help 
elderly nursing home residents become more alert and care fill of their medications. This 
encounter provided a natural opportunity for Karen to experience Ann's positive support, 
an opportunity Karen would not have sought on her own. 

Karen did better on the second exam but not as well as she hoped. When I 
commented that her 76% was a lot better than her 62% on Exam #1 she demurred, 
saying, "But they were so easy, the ones I missed." Her focus seemed to remain on the 
negative. Unlike the first exam when her formula sheet had adequate column prompts for 
formulae such as: 



X 


x-x 


(X-X)^ 









to prompt the correct use of the sample standard deviation formula: 



(x-xY 

n-l 



223 

that she failed to take advantage of. this time her formula sheet did not have a necessary 
column prompt so she failed to compute a statistic correctly."" Karen interpreted this error 
as a procedural rather than conceptual failure on her part, but her failure to use her correct 
formula as a prompt for the missing column indicated that she had not explicitly linked 
the formula with each procedure she needed to follow — reasonable given her low level 
understanding of the algebraic variable and the fact that this was the problem we had not 
got to in Session 2 and that she had not therefore practiced. She had skipped another 
question because she had not understood what the question was asking. Karen's strategic 
preparation had improved but not sufficiently to compensate for her algebraic weaknesses 
and because of this lack of preparation, her formula sheet was inadequate. 

Karen was reporting on her exam, as it was not available (students returned exams 
to Ann once they had looked at them — Karen had not arranged with Ann for her to give 
the exam to me for our session.). Karen did not mention the complex questions I saw later 
she had done and interpreted correctly on the exam. She had 78% of the computational 
section correct — a significant improvement on her 52% on this section in Exam #1 and 
on material that was mathematically and conceptually more complex. 

At my suggestion we worked on an inference test problem worksheet Ann gave 
out for students to try on their own in Class 1 1 (Mann- Whitney U test of separation of 
two populations). The course focus had shifted from descriptive to inferential statistics 
and Karen had grasped the strategic importance of following the same hypothesis testing 
procedures for each test (using the step by step hypothesis testing procedure list Arm had 
distributed, see Appendix G). Again I used parallel tutoring and diagrams to aid 



224 

conceptualization and Karen struggled successfully creating a careful organizing scheme 
to provide structure. 

I noticed that Karen was fast and accurate at addition of whole numbers, 
something I am not good at: 

JK: ... they want you to add up the rank. [Karen circles the ranks that the c- 
group got] there you go, there you go; there you go; right now add those 
up 

Karen: 16 ([immediately] 

JK: Oh good on you! You did that boom! Wow quick! (Session 3) 

And I found more and more opportunities to help her notice how she could move 
out of cloudy misunderstanding into the clear light. For example, after Karen had begun 
to add ranked scores instead of ranks we looked at them more closely and she saw it: 

JK: so what is the difference? What is the difference between a score and a 

rank? 
Karen: Well this is just how, um, chronologically where each one falls and that's 

the score that each one got {exactly} how many numbers, I mean how 

many words they actually remembered 
JK: Right, right and they have them ranked there because you put them in 

order {right} but they are still each one are scores so the "rank" there is 

almost an adjective {right} okay? {rightt} 
**Time (1-2 seconds) as K adds up 
Karen: I can't subtract worth a dime // 
JK: But you're adding is like whoosh, right? (Session 3) 

Again she added a string of numbers almost instantaneously, again drawing my 
admiration despite her negative comment about her subtraction abilities, seemingly 
designed to keep my attention on her deficits. She seemed unused to receiving positive 
recognition for her mathematical work. 

I continued to feel that Karen was maintaining emotional distance from me. In the 
following interchange she seemed to be carefully considering each word so as to reveal a 



225 

minimum of information and she ended the exchange by abruptly turning back to the 
statistics. 

Karen: Usually I have this all done. I've been a little too harried 

JK: oh^ oh! Other things going on in your life? 

Karen: Well, no, Tve just been **, um, *** away. 

JK: Away? (giggle) I don't know; do you find Fourth of July throws things off 
a little bit {yeah = yeah=} oh, vacation! {yeptjand all of a sudden you 
remember it's summer {mmm} and you think summer we're really not 
meant to be doing this {right} is that what happens to you? {Yep} So 
where'd you go? 

Karen: Um I just went to visit *some friends {yeah} but um* Okay, so the ranked 
score 

Our interchanges about her former mathematics teachers and her family's 
mathematics orientation were similarly limited. I was experiencing this as her not feeling 
secure enough to trust me further and that seemed to have been a long established pattern 
for her, at least with mathematics teachers. 

At this session, because Karen seemed less stressed and more focused than in 
Session 2 and she had done better on her second exam, I expected her responses to the 
JMK Scales, to have become more positive (see Figure K3, responses labeled 3). 
However there were only small positive changes on items 1 , 2, and 4, her sense of 
hopelessness about her mathematics future had increased, and the other responses 
remained the same. From Karen's point of view, it seemed her short-term fliture in this 
class was not assured. She had made gains on Exam #2, but her control of the material 
still felt uncertain. The last time she took this class, she had improved on the second 
exam, too, but it had not been enough to pass. A D would not be sufficient; for her major 
Karen needed, at minimum, a C. She was taking responsibility for doing more, but her 
focus remained on mastering procedures, rather than concepts. She was no longer 
blaming Ann, but now her self-criticism seemed, perhaps, too harsh. 



226 

Exam #3 was scheduled for the following Monday. Karen asked for an extra 
appointment before the exam. This request indicated to me her growing sense of 
responsibility and her ability to take some control by getting the help she needed when 
she needed it. I wondered if it also signified that the strategic urgency she felt was 
sufficient to override her defensive distancing from me at least with respect to the 
mathematics. 

Karen 's Session 4 

Karen came to drop-in several hours before our scheduled appointment. I was 
anxious for the students, especially Karen, on Exam #3 because it was the first one on 
inferential statistics and hypothesis testing, and because of the number, variety, and 
complexity of the inference tests to be examined (see chapter 5, discussion of Class 13, p. 
175 and Exam #3, p. 179). 

I offered Karen the materials I had prepared (see chapter 5, p. 179) and she 
decided to use the flow-chart template to create her formula sheet for the exam.'"' 
Karen's other organizer was her "Steps for Hypothesis Testing" list. During this drop-in 
session, 1 was also working with other students, but I checked with Karen from time to 
time to discuss and help her work through her struggles in deciding which test to use for a 
particular situation. Deciding between the two-sample mdependent / test and the two- 
treatment correlated t test was difficult for her so we discussed ways to decide which test 
to use based on the situation described in the question. 

At our scheduled meeting time. Karen had completed three of the six problems I 
had given her, with only occasional help from me. I commented that she was much 
calmer than she had been before the previous exams. She agreed to some extent but 



227 

qualified it. She seemed vulnerable to anything that went outside the structures she was 

carefully building for herself. 

You know if she changed a word or the order of the words just one little bit, I 
wouldn't know what to do... that's what happened to me in the last test. She had 
worded it differently so I sat there and looked at it [what I'd done] and I took it 
out to her and she wouldn't say. . .1 was so confused. . .1 just left it because I had to 
go on to the next one. (Session 4) 

I asked her what statistical test the next question called for. She decided on an 
independent t test and I asked about her reasons. She hesitated and asked for time to look 
at the question, then said firmly, "It is an independent t test because she has given us the 
two s- s." I affirmed her choice and her thinking. In retrospect my response surprised me. 
This was quite different from my response to similar reasoning by Mulder. When he used 
this reasoning I remonstrated, insisting that he link his decisions with the logic of the 
setting by determining whether they originated from two different groups of subjects or 
from one group of subjects tested twice. 

Why did I not do that with Karen? I had had that discussion with her during drop- 
in earlier but I didn't bring it up again now. I think I was responding to my sense that she 
was carefiilly building up a fragile personal structure for negotiating the exam that I 
hesitated to challenge too forcibly."^" I did not want to risk upsetting the procedural 
control she seemed to be gaining over the material for the imminent exam by pushing her 
to make these logical links. 

For the remainder of the session Karen worked problems while she carefully 
organized her formula sheet and I quizzed her on the definitions and the sample versus 
population categorizations of each letter symbol. By the end of our session Karen had 
completed at least one problem for each of the inference tests being examined. 



228 

I noticed out loud Karen's impressive grasp of the material and at first, she denied 

it, attributing her success to external factors such as being in mathematics counseling not 

in the exam room or "cheat[ing] in the book." I challenged that thinking pointing out that 

it was not me helping her; she was doing it herself. And she challenged herself noting 

that she hadn 't used the book only her formula sheet that she could use in the exam. 

Karen: So I just know that this and that are the same thing! 

JK: Right {alright (+)}. You are doing quite sophisticated mathematical 

thinking!! 
Karen: Yeah, but when it comes time for the test I'm not going to remember it (-). 

Maybe if I look over it just before. 
JK: You will. You are able to do this, you know. I'm not helping you at all 

you're just doing it yourself 
Karen: See this is going to be my problem. This was already done you know 

what I mean? I cheated in the book. No, well I just looked at this [decision 

chart formula sheet that she had been adding to and could use during the 

exam] actually 
JK: You really did and you'll have this on the test. Right? And you've become 

aware of how you might be tempted to choose one [statistical test] rather 

than another! Right? So I think you probably won't, right? 
Karen: 'Cause I could probably get halfway through the problem and realize {I 

would think so} that it wasn't right (Session 4) 

Karen went on. She realized she had now intemaUzed the material and ways of 
troubleshooting on the exam if she got into trouble. It was remarkable change from how 
she had experienced her confiision during Exam #2. Karen was going into Exam #3 a 
very different person from how she had gone into Exam #1 . She had prepared 
strategically because she was comfortably aware of what would be tested; she could 
compensate for her algebraic and arithmetic deficits; she had a carefiilly prepared and 
strategic formula sheet, and now she had become aware of her own grasp of the material 
and her ability to monitor and troubleshoot if things went wrong. 



229 

Discussion of Sessions 3 and 4 and Exam #3 
My Relational Focus in Counseling 

Karen's responses to the JMK Mathematics Affect Scales in Session 4 before 
Exam #3 revealed that she felt considerably more able to make mathematical decisions 
and she was more positive about herself mathematically than she was for the previous 
exams (in Appendix K, Figure K3, compare responses labeled 4 with those labeled 2). On 
the other hand, the material was much more complex, and Karen seemed to be relying 
heavily on extrinsic clues and her formula sheet. Overall, her improved sense of 
mathematical efficacy (items 5, 7) seemed to be breaking up her sense of hopelessness 
(item 3), while the uncertainty of the imminent exam and her reliance on procedures and 
extrinsic artifacts seemed to have kept her discouragement (items 2, 4, 6) from 
dissipating. 

Karen earned 85% on Exam #3 and she was very pleased. She lost only 2 
points'"'" on the computational part, 12 points (6 questions) on the multiple-choice and 1 
point on the symbol identification. As noted in chapter 5, Karen was one of only two 
students in the class whose grades improved on this test. With the in-class extra credit 
assignment where she earned 6 points, her overall exam grade was 91%. 

Even though Karen was gaining control where she previously had felt helpless 
and discouraged, even depressed, and her mathematics depression appeared to be lifting 
somewhat, I was concerned that she had not made an extra appointment to address her 
underlying number and operation sense weaknesses. I realized that her teacher relational 
detachment issues would have made it very difficult to seek that appointment. It had 
likely been difficult enough for her to make the extra appointment before Exam #3. 



230 

Supervision 

In presenting Karen at my own supervision session with Dr. P., I expressed a 
history of my thinking about the changes she had made that I realized, as Dr. P. heard, 
were considerable. "She's doing better and she is growing into a competency she didn't 
know she had." 

Dr P. suggested that Karen might have a new metaphor for herself doing 
mathematics, and that it could be helpfiil for her to assess her own change. At each 
session, I had been prompting Karen to look at what she was doing differently and seeing 
differently. He suggested that reflection should continue at the next meeting. I wondered 
how to help her reflect more deeply. Her underlying arithmetical and mathematics issues 
and our limited opportunity to explore and discover her real ability to think conceptually, 
dampened my enthusiasm about her current success. Was her success "good-enough?" 
For this class, perhaps! But I worried about its strength for restoring a healthy 
mathematics self Karen's opportunity for reflection would come when she did the final 
evaluations and posttests. 

Karen 's Session 5 

Again Karen came early to drop-in, this time with a little grin and feeling "good" 
about her grade on Exam # 3. She told me that her family were "waiting at the door" for 
her, obviously very pleased. Then she began doing homework problems from the text. 

In the individual session, Karen tackled a question where she was asked to fmd 
the values missing from a one-way independent-groups ANOVA summary chart and 
answer questions about it. She was stuck on the question, "How many groups are there in 
the experiment?" (Pagano, 1998, p. 378). She said, "I don't know what to do." This was 



231 

the type of inverse reasoning question that Karen found difficuh. I suggested some ways 
to think about how those numbers were derived and what they meant and I coached her to 
think backwards to find the number of groups and, in the next question, the number of 
people in each group (assuming equal numbers in each group). 

Karen and I discussed strategy for preparing for the exams. In response to my 
query about completing homework problems prior to coming to the Learning Assistance 
Center, Karen replied, "Ah no, I did those tests [problems] that you gave me. They were 
way more helpful than doing all this homework." 

Karen had never directly attributed her difficulties in mathematics to lack of 
ability. In the Beliefs Survey she had agreed that her ability to do mathematic could 
improve but she also agreed that some people can do mathematics and other people can't. 
In addition she labeled herself (and her family) as a reading and writing type [not a 
mathematics type]. I took the opportunity in the following discussion to bring up the 
topic of ability. 

Karen: Right, I was not interested in math at all. I don't like it; that's why I don't 

do as well 
JK: Yeah that's got a lot to do with it probably, not much to do with abilityt 
Karen: Probably not 

JK: More to do with interest and CONFIDENCE 
Karen: Right 
JK: Because as you don't do as well, your confidence goes down {exactly} 

You THINK you're not good at it (Session 5) 

Karen's attributing her not doing "as well" to her lack of interest sounded 
defensive. When I mentioned ability as a possibility and dismissed it she gave only 
qualified assent "probably not." That was when I told her of Liping Ma's (1999) work 
studying American elementary teachers' generally poor grasp of arithmetic. I speculated 
with her that she most likely had teachers who themselves had not understood any more 



232 

than how to do the procedures. She agreed. I could tell that it would take more than her 
good grade, my mirroring of her competencies, my logic, and research findings to 
convince her of her sound ability to do mathematics, but all of these were making rather 
large chinks in her armor. 

Karen was pleased when I noted approvingly that she continued to follow the 
hypothesis testing procedure meticulously. She commented that, in class, the other 
students weren't doing it, but she was. Karen was beginning to recognize that more of her 
mathematical behaviors were positive, in dramatic contrast with her former almost 
exclusively negative evaluation of herself doing mathematics. 

We worked on an ANOVA together. At one point I had a formula incorrect but 

Karen had it correct so our answers were different. I questioned her but she held her 

ground and then I realized that it was I who had it wrong. 

Karen: I get more than that as the first one 

JK: Why are you squaring that again? 

Karen: I've just got the sum of x-one [X|] 

JK: Oh you've got the formula copied wrong [inspecting Karen's work] 

JK: Oh no, you don't! I'm doing it wrong. You're doing it correctly! 

Karen: They're really big numbers though. (Session 5) 

Karen's caring attempt to reassure me that my mistake was understandable: 

"They're really big numbers though," marked a reversal. Karen had experienced tolerable 

disappointment in me, the idealized teacher-parent, at the same time as she realized that 

she had it right. Her competent self was emerging and could care for me the parent. 

During class problem-working sessions, Karen continued to show no interest in working 

with anyone other than "experts" in the mathematics, in this case Aim or me. While the 

parent image was still prominent it was being modified by reality and incorporated into 

her mathematics values structure. 



233 

And Karen had changed her mind about Ann. Now she recognized Ann's efforts 
on her behalf and her defensive detachment had been replaced by a sense of secure 
attachment, as illustrated in the following exchange. 

JK Oh you'll plug all those into this. I can't imagine trying to do- 1 know it's- 
I hope she gives us one with a lot smaller numbers [on the exam], that 
would be better but no matter what 

Karen: She usually does. {She does} Even when she gives us the practice 

problems she never has the [large number of large numbers]. I mean, the 
book is ridiculous sometimes like these aren't the biggest charts I've seen 
like way back when we were doing just frequency distributions like a 
whole page was writing; it was really long. 

JK: Yeah, that's right it was wild. I think they do that because 

Karen: She even made a comment too she said 'i'll never make them as long as 
the book does." (Session 5) 

It was in this session that I noted too a distinct change in Karen's emotional 
distancing from me. When I commented on different national views on mathematics 
ability and I mentioned Austraha's, Karen talked at length of her girlfriend's visit to 
Australia. 

As with Session 2 we had not covered all that would be on the exam. In fact, even 
more would be covered in class tonight that would be on the Wednesday exam. But I was 
confident that Karen had it well in hand and she was too. Although I did not know it at 
the time, this was my last meeting one-on-one with Karen. 

Karen 's Post Counseling Processes 

Karen earned 88% on Exam #4. She was very satisfied. She lost no points on the 
computational and symbol identification sections. All her points were lost on the 
multiple-choice (6 out of 23 questions incorrect — a consistent result; see Appendix K, 
Table K2). 



234 

After Session #5 we had scheduled an appointment for the following week that 
Karen cancelled. I was concerned that we meet before Exam #5 because I knew the exam 
would require students to decide on an inferential test using a specific decision flow 
chart. I knew the questions on the exam would not contain the specific clues I thought 
Karen might be relying on such as the '5- 's for the independent samples t test so I 
suggested she come to drop-in on the day of the exam; she did not come. 

In our final session her self-reliance had been remarkable. After that she felt she 
could handle the rest herself, and she did. I struggled with my countertransferential 
parental concern. It was hard for me to let her go and trust that she was in a good-enough 
place, that she could do it on her own but I need not have been concerned. In fact, I 
should have been pleased at Karen's growth. She earned a 96% on that exam (although 
she did fail to correctly identify the independent samples / test!). 
Optional Comprehensive Final 

Karen decided to take the optional comprehensive final after class ended to 
replace her lowest exam grade. I offered an appointment by e-mail, but she declined, 
which made me quite anxious for her again. This was not my countertransference alone. 
Students' grades on comprehensive mathematics exams, even with review, are typically 
one-half to one whole grade lower than on their other tests. Karen earned a 57%, which 
was lower than her lowest test grade so it did not alter her final grade, a B. She had badly 
failed the conceptual multiple-choice part but on the computational part of this test Karen 
earned a 75%. Although this 75% was considerably lower than she had been getting on 
computational sections, given that the exam was comprehensive, and that she took it with 
no class or tutorial review, it was reasonable for her. Even her overall poor result was 



235 

relatively comparable with results of others who took the comprehensive final (see 
chapter 5, Table 5.1) and unlike her each of them had had a fmal review session with me. 
I wondered whether, however, without discussion of this overall low grade Karen might 
allow it to dimmish in her mind the real gains she had made in her mathematical prowess 
(Appendix K, Table K2). 

Evaluations 
Karen 's Evaluation of Her Changes 

Karen said her initial "cloudy" metaphor for herself doing mathematics may have 
changed "a little" but she was not specific. She predicted she would not "ever like math" 
but that she was "more comfortable" with it. Karen attributed her own positive changes to 
"1 on 1" and to the "amount of time I put in outside the class" (One-On-One Mathematics 
Counseling Evaluation). She learned that she could "do a lot better than I thought" but 
still found the "conceptual" aspects of statistics puzzlmg and would "pay particular 
attention to the conceptual portions" of the next mathematics-related course she took. 
My Evaluation of Karen 's Changes 

Karen took the Feelings and Beliefs posttests in class, and the Algebra Test and 
Arithmetic for Statistics assessment after she had taken the optional comprehensive final. 
These two tests confirmed my sense of Karen's weak arithmetical and algebraic 
understanding (see Tables 6. land 6.2, respectively) and they also confirmed her need for 
compensatory structures and strategies to achieve the success she did. 

By the end of the course Karen's overall defensive and detached pattern of 
relationship in the classroom setting (or possibly the college setting) had eased. She 
engaged in conversation with other students during the problem-working sessions but still 



236 

she would not work with or check her work with anyone but Ann or me. Her initial 
angry negativity towards Ann had reversed. She had begun to forge secure attachments 
with trustworthy mathematics teachers — Arm and me. 

Karen's sense of herself as a mathematics learner had become a little more 
positive. By the end of her last individual session Karen's discouragement responses 
(items 2 and 5) on the JMK Mathematics Affect Scales had lifted (see Figure K3, 
responses labeled 5). She also indicated that she was less likely to withdraw from the 
course. Until Session 4 (just before Exam # 3), Karen's responses on all scales were at, 
spanned, or fell below the mid-points. Now at Session 5, in 3 out of the 7 scales, her 
responses were above the midpoint (positive) and the others at least touched the 
midpoint. 

Karen's moderate mathematics depression had lifted somewhat in the context of 
the mathematics counseling and the current course. However, her mathematics depression 
appeared to have developed over many years of school mathematics in an envirormient 
focused on procedural mathematics learning which she had little hope of understanding 
and where her developmental needs were neglected. This resulted in an underdeveloped 
mathematics self: she was underprepared mathematically and her mathematics self- 
esteem was therefore low. Ann's course forced her to tackle procedures on her own, and 
a formula sheet was allowed, so it was possible for Karen to gain control and succeed. 
She developed a more positive sense of her mathematics self and moved from an 
unhealthy detached independence to good-enough mathematical self-reliance. For a 
lasting improvement and success in a more conceptual mathematics course, I believe 
Karen would have to understand arithmetic better and develop her understanding of the 



237 

algebraic variable. She still had finite mathematics ahead of her, and she planned to take 
it in summer 2001 . She said she would return to the Learning Resource Center for 
support "as long as Jillian is there" {Follow-up e-mail Survey). I worried that might not 
be enough. 

The changes in Karen's responses on the post-course Mathematics Feelings 
survey (see Appendix K, Figure Kl) and Mathematics Beliefs survey (see Figure K2) 
surveys seemed largely consistent with her changes and her success in the course, 
although there were some apparent anomalies: Although her abstraction and number 
anxieties had decreased substantially, her testing anxiety had increased (see Appendix H, 
Table H3). 
Evaluation of Counseling and My Changes 

When I met Karen I was immediately drawn into her anxious, depressed, 
negativity. With her I saw her deficits and limits and heard her anger at Ann and 
despaired of her making it and of my being able to help her. But as I incorporated 
relational counseling assessments and approaches into best practice modified cognitive 
constructivist tutoring and course management counseling, I changed my mind about 
Karen and about me. As I helped her see Arm and herself differently I began to see her 
differently. My expectations of her rose, my role changed from motherly rescuing to 
guiding hand and co-explorer and she rose to the occasion. We found ways for her to 
compensate for her significant background deficits and my admiration of her grew. Going 
beyond tutoring to incorporate relational approaches led to her not only doing the 
mathematics but also to her recognizing herself doing the mathematics, and her 
underdeveloped mathematics self developed. I (and Ann) had provided the opportunity 



238 

for her to forge secure attachments to mathematics teachers and she had availed herself of 
that opportunity 
Evaluation Summary 

Karen's mathematical relationship patterns had begun to change. Her mathematics 
self was becoming firmer; she found she could gain control over the mathematical 
material to a greater extent than she had ever thought possible. I felt she was still quite 
dependent on teacher/tutor input and external judgment of her mathematical correctness 
rather than on her ability to judge the internal consistency and logic of the mathematics. 
But once she worked out how to use structure and strategic effort to compensate for her 
mathematical uncertainties, she did it on her own. Her final reflections indicated the 
movement she had made towards an improved sense of her mathematical self and 
mathematical self-reliance: She wrote, "I became more confident as the course went on 
and I came [to drop-in and individual mathematics counseling sessions] more for security 
in knowing I got the answers right" {One-On-One Mathematics Counseling Evaluation). 
She apparently felt she had enough of a mathematical self to do it herself; she no longer 
needed me except to check that she was on the right track. 

Epilogue 

Karen did enroll in Finite Math in the summer of 2001 and she did come to the 
Learning Assistance Center to get help fi-om me. Following her pattern of summer 2000, 
she came first just before her first exam, overwhelmed with the amount of work, resentful 
that her transitional object — a formula and procedure sheet — was not allowed, and not 
having practiced each type of problem. The instructor allowed her an extra few days but 
she still did very badly. Karen regrouped and began to come regularly to Drop-In. She 



239 

did not like my going from person to person at Drop-In and not attending solely to her so 
she began to work on her own in the cafe and would come down to the Learning 
Assistance Center during Drop-In just to ask specific questions and then go away again. I 
suggested we meet to deal with her arithmetic issues, which surfaced again but she never 
made that appointment. 

Karen's mathematical self-doubt remained a problem: Although she felt confident 
with Venn diagram counting questions, on a take-home quiz she erased and changed her 
answers when another student had different ones only to fmd out later that she had been 
correct. "I always assume that I am the one that is wrong." Her belief that the teacher was 
against her also returned though I did not feel included in that this time. As I suspected 
her mathematics depression had deepened again since the end of PSYC/STAT 104 but 
she persevered, and I continued to confront her globalizing self-negatives with proof of 
their fallacy from her own work. Again, her grades improved. She made and kept two 
individual appointments before the fmal, when I was more able to take a mathematics 
counseling approach with insights from our earlier counseling sessions. Karen was 
organized and knew what she needed to learn. She was allowed to use a restricted 
teacher-developed formula sheet and went on to earn a B' on the cumulative final and a 
C^ on the course. This was quite an achievement because it was a more mathematically 
demanding course than PSYC/STAT 104. 

For Karen this was a good-enough outcome. All the mathematics requirements for 
her degree were completed. She will probably not take up the challenge of dealing with 
her underlying operation sense, number sense and algebraic deficits, which are at the root 
of her mathematics depression. 



240 

JAMIE'S COURSE OF COUNSELING 
Jamie needed help with her statistics course. She decided so herself. I know this 
because she signed up for mathematics counseling with me for once a week, not once 
every other week, which was an option, and later, in her end-of-the-course evaluation, 
she wrote that her initial motivation for signing up for counseling was "so that 1 could get 
a better grade in the course," unlike other participants whose initial motivation was to 
help me with my research""^ (cf Mulder, Robin, and Autumn). But if I had not crossed 
lines with Jamie that are generally drawn in the helping professions, it is unlikely that we 
would have worked together at all. As a helping professional I had learned that I should 
wait for the person seeking help to approach me; it is usually considered unacceptable to 
pursue the student in order to provide help, no matter how necessary that help seems to 
be. Jamie, however, despite signing up for weekly counseling sessions and despite an e- 
mail exchange between us about when, slipped quietly away after class night after night 
until finally I decided to sit beside her in class in order to arrange the appointment she 
had indicated she wanted. 

Jamie was a tall, dark-haired, white, traditional-aged fiill-time student at State 
University who had just completed her sophomore year. Her father was an engineer and 
her mother was also college-educated. As a psychology major, Jamie needed 
PSYC/STAT104'" but thought it might be easier to do it here at Brookwood State in the 
summer; the small class size and focus on only one course, she thought, should more than 
compensate for the course being faster than in a regular semester (ten weeks compared 
with 1 5 weeks to cover the same material). Jamie had withdrawn fi-om Finite Math in the 
fall of 1999, without penalty because of illness, although she was failing at the time 



241 

(Session 1 ), so the last mathematics course she reported that she completed was pre- 
calculus in high school in which she earned a "C?'""' {Pre-Test Mathematics History 
Survey, see Appendix C). Ann, the instructor, thought she was "VERY quiet" and used 
the word "fragile" to describe her (Interview 2). Jamie wrote that she hoped for a B in 
PSYC/STAT 104 but expected a C {Pre-Test Mathematics History Survey). Her summer 
job was in a department store in a mall. 

What struck me most about Jamie at the first class was her demeanor — she was 
sitting straight up with her eyes lowered. At times I wasn't sure if she was asleep but her 
expression did not seem to change and she did not make eye contact or interact with 
anyone, except during the paired introductions interview when she told her interviewer 
that she was "not keen" on mathematics or doing this course. 

I found out that Jamie was cognitively capable and well-enough prepared 
mathematically to succeed, yet in two attempts at mathematics courses in college she had 
not succeeded. Jamie's personal and mathematical style and challenges induced her to 
accept my offer of help but dissuaded her from accessing it. And hers contrasted 
markedly with my personal and mathematics style and challenges. Mine induced me to 
cross accepted helper boundaries to give her the help she needed but caused me to 
struggle with helping her fmd her voice when mine was so loud and hers so quiet. How 
we understood and struggled with, negotiated, and made use of our differences together 
forms the substance of this account of Jamie's and my growth as tutee and tutor over the 
summer of 2000. As I used the relational counseling approach that I delineated m 
chapters 2 and 3, 1 looked at her and at myself differently from how I would have in my 
former practice. Both Jamie and I benefited — she "realized it was more about my feelings 



242 

and confidence in my math ability, th[a]n any real problems with the math course work" 
and she earned a B^ and I learned how attending to our relationship helped me understand 
her and myself better and modify my approach with a student who was so different from 
me. 

Jamie 's Metaphor: Mathematics as Stormy 

By the time Jamie and I met for the first time in the fourth week of the course, she 
had received the results of the fu-st exam and to her dehght and surprise, had scored a 
95%. Nevertheless, her metaphor for mathematics was a [violent] thunderstorm. She 
explained her choice: "stormy because it is usually very tough for me to do and 
imderstand math, even though I did good on the test I'm afraid the 'storm' will come 
back again" (archived College Learning Metaphor, see also Appendix B). 

For a storm, Jamie said, she would, "prepare for it; before it comes, like, get your 
water or flashlights." When I asked how she would handle the storm when it came, Jamie 
replied that she would "stay inside." She saw how her storm preparation related to 
mathematics: "Well, you have to prepare for tests," but she wondered "how staying 
inside does." We did not initially explore what the storm itself was to Jamie — I assumed 
it was mathematics itself, in particular, mathematics tests. I did not pick up then on the 
connections between one of her other metaphors "shark," her use of the word "afraid" in 
her "stormy" metaphor, and the link to my countertransference experience in the first 
study group: my experiencing being potentially dangerous to Jamie (see chapter 5, Study 
Group 1). I also didn't attend to her wondering what "staying inside" out of the storm 
might have to do with her doing mathematics. 



243 

Over the course of the first three meetings Jamie told me of her stormy 

experiences with previous mathematics classes. Her most recent experience, she told me, 

was withdrawing failing from a finite mathematics class at State University'"" and high 

school had been mixed. The storms began in elementary school, however. 

Student-teacher Relationships as Stormy: Jamie 's Internalized Teacher Presences and 
her Mathematics Self 

Jamie's early elementary experience of mathematics sounded calm: "first grade 

and second grade and stuff, you know, I got 'A's in everything," and she remembered 

she'd liked her fourth grade teacher. Her experience of 5"^ grade had been different: Her 

5* grade teacher "yelled" though not at her, and not particularly about mathematics. 

Jamie attributed the start of her doing poorly in mathematics and science to the 

frightening classroom situation tills 5* grade teacher created, though her reading and 

writing achievement remained unscathed. She remembered: 

But in 5th grade, my teacher kind of yelled a lot, and stuff, and I didn't do good 
[in mathematics] . . . Science, I think, too. . . I did good in writing and reading, that 
kind of stuff. . . It was from then on. . .1 think she had a short temper, I guess. 
(Session 1 , June 20) 

In Session 3, as I was asking Jamie about her shy, non-interactive demeanor in 
Aim's class, the effect of her 5' grade teacher came up again. Jamie explained further 
"You want to sit down and shut up so you don't bother her [the 5* grade teacher]." I was 
struck with how closely this described Jamie's current behavior that I observed in class. I 
was also aware of how much Ann, the instructor's, approach differed from Jamie's 
description of this 5'^ grade teacher. 



244 

In high school, to Jamie's surprise ("because I don't do good in math") she "did 

good" (a B or B^) in Algebra I. The storm hit again, though, in precalculus that she took 

with the same teacher she had for Algebra I. Her experience in precalculus was so "bad" 

that by the end she said she didn't understand anything and she found the teacher to be 

"stand-offish, like, 'You should know this.'" She remembered needing little help in 

Algebra I. It had gone smoothly ("I didn't do bad and good and I wasn't up and down"), 

but when she did need help in pre-calculus, she (and the other students, she said) found 

the teacher to be unavailable. Jamie conceded though, "Well, part of that not getting help 

is partly me." Jamie's unwillingness to seek help from the teacher (i.e., Jamie 's 

unavailability), she believed, contributed to her problem of not getting the needed help. It 

seemed to me, however, from what she said that she had been inhibited, not only by her 

"stay inside" relational pattern, but also by her observations of other students' difficulty 

in getting a response from the teacher. She perceived this to constitute a negative change 

in the teacher from her Algebra I experience of her. Mathematics teachers had become 

potentially dangerous to her. It was as if she had internalized bad mathematics teacher 

presences through whom she saw Ann and me or any mathematics teacher. And all her 

difficulties she attributed to her own inability to do mathematics. 

Student-mathematics Relationship as Stormy: Attachment 
and Jamie 's Mathematics Self 

I reflected on the probable effects of Jamie's stormy history on her sense of 

mathematics self Her attachment to mathematics and to mathematics teachers had been 

secure through fourth grade. Then in fifth grade her expected secure teacher base was 

withdrawn: She could no longer safely explore and ask for or expect the support she 

needed. Thus began her sense of isolation, separation from a secure teacher base and. 



245 

from then on, from a secure base in mathematics — she could no longer be sure that she 
understood it, sometimes she did well, other times she did not, but she could not ask why 
because she was no longer sure of the availability of the teacher. 
Jamie and Me — Dealing with Storms Now: Relational Counseling for Jamie 

As I reflected on what "stormy" meant to Jamie, these understandings clarified for 
me the effects in this class of her current expectation and fear of these storms continuing. 
Our differences became more apparent but I also became more aware of what I needed to 
be for her. "Stormy" seemed to have multiple meanings to her, all negative, but the 
consistent theme was absence of calm — teacher "yelling" or "ups and downs" in 
understanding or grades. I, on the other hand, enjoy storms, especially the thunder and 
lightning, and calmness bores me. I would have liked to persuade Jamie that "stormy," 
like mathematics, might have positive aspects — challenge, excitement, darkness lit up by 
the lightning. As — but I gradually realized that none of these (probably including 'A's) 
would feel agreeable to Jamie. If I could offer myself as a smooth, level path with no 
surprises around the comer, only more of the same, or perhaps a gradual ascent, nothing 
that would startle her or trip her up, that would be perfect. 

Jamie 's Mathematics Relationality 
Interpersonal Relationships and Self: Family and Personality Interacting 

At Jamie's second session, just after she had taken Exam #2 but before she 
learned her grade, I asked about her family's reaction to her 95% on the first test. Jamie's 
Dad was pleased and had expected it to continue; her grade was proof to him that she 
could do well in mathematics. Jamie saw it differently; this was not proof but rather an 



246 

anomaly, not likely to be replicated. She knew she had not done as well on Exam #2 and 
she was not surprised. 

She had negotiated her panic on Exam #1 when she found and corrected an error 
so I presumed that that success and her high grade would result in reduced anxiety for the 
next exam. On the contrary, Jamie said she had higher anxiety on the second exam 
because of her family's (in particular, her father's) higher expectations. I realized that I 
had to navigate my own assumptions and expectations of Jamie. As I proposed a 
conjecture and learned to listen to Jamie's responses, including her hesitations, 
qualifications, and tone of voice, she changed my mind and revealed herself Although 
she never contradicted my conjectures about her, Jamie's unconvinced "maybe"s 
contrasted with interested and curious "possibly"s; her hesitant "yeah4'"s contrasted with 
her somewhat "yeah="s and her firm, in-agreement "yeaht"s, laughs, and "I know"s. 
Jamie usually qualified her own theories with "I guess"s and "maybe"s but I had to watch 
and listen to clues to how deeply she held these theories (See chapter 4, Table 4.3 for 
transcription coding conventions I use.). 

JK: And the anxiety in the second was just to do with that confusion about the 
~ [Jamie had just told me that she had known how to do the various 
correlation and regression computations on the exam but had been 
confused about what each one were called.] 

Jamie: Yeah=, that and I think I might have been more [anxiety] for the second, 
actually 

JK: You were more anxious on the second one? 

Jamie: Yeah, I think so. 

JK: That's interesting, (surprised) Does that happen to you? Like for the first 
one in your course you're not quite as anxious? 

Jamie: Maybe>l' (unconvinced) 

JK: Why do you think you were more anxious for the second one? 

*Time* [here I waited for Jamie to answer — several seconds] 

Jamie: Umm, Well, I know this time why I was. 

JK: Okay, why? 

Jamie: It was because of my 95! !! 



247 



JK: Ahhh! That's interesting! That made you more nervous? Now why? 
Jamie: Weil, I guess 'cause my parents {Alih!} were expecting it to be maybe a 

similar grade. 
JK: Oh, so there was this high grade and it was really possible not to get that? 

{yeah t} put a lot of pressure on you? So do you think the actual level of 

confusion [also] contributed to your conflision? 
Jamie: Possibly= 
JK: Shouldn't tell them [your parents] your grades... {(laugh) I knowt} keep 

that to the end, but you were so excited it would be hard to keep that to the 

end... 
Jamie: Yes. (Session 2, July 3) 

When I offered the suggestion that Jamie might be relieved to get a lower grade 

on the second exam I didn't feel as if I was putting words in her mouth and her strong 

"YeahT"s confirmed this. 

JK: So it actually may be a relief to get a little bit of a lower grade? 

Jamie: Yeah t 

JK: And then you won't feel so much pressure on you for the next one. 

Jamie: Yeah t (Session 2) 

She seemed to have experienced the 95% grade as much as a storm as she might 
have a really low grade, an "up" that she seemed to dread as much as a "down" — ^the 
absence of calm. And I was surprised and curious. How could this be? It was hard for me 
to entertain the possibility that an A might constitute a burden for someone. When 1 
considered where I stood in Jamie's world, I had been more with her parents than with 
her, not only in my own mind but also perhaps in hers. I heard her conflicting 
motivations — to "do better on the course" but also to maintain calm, that is, not to do too 
much better, not to raise hopes, not to elicit external pressure to maintain to her, an 
impossible standard. As she explained herself in contrast with her parents, I became more 
aware though, of how my expectations of her might differ not only from hers but also in 
some ways from her parent's. Could I hold high expectations of her without exerting the 
accompanying pressure that made her so anxious? Yes, I decided, because, unlike 
Jamie's parents, I was positioned to be able to help Jamie explore to what extent these 
expectations were realistic and to own them for herself if they were. 



248 



In the next session (3), I asked about her parent's reaction to the 74% she had 
earned on Exam #2. 

Jamie: Um, I don't know. I guess my dad was just kind of Hke, "Why did you get 

a 74?" or something. 
JK: Really? {Yeah} Especially when you got the 95, right? And what did you 

say? 
Jamie: ...I just kind of said, "To me it was more surprising that I got the 95 than 

the 74." You know? (Session 3, July 1 1). 

Jamie seemed calmer; as her parent's expectations had been reduced so also was 
the pressure. Jamie and I could continue to explore and challenge her expectations with 
evidence of her prowess and achievements. 

I used the word "quiet" for her when we discussed her reaction to the 5* grade 
teacher and her demeanor in class; it was held to be self-evident in our discussions. Jamie 
agreed that she was quiet, like her Dad. During the course, she never used words like 
"shy" about herself although in the Follow-up E-mail Survey she did. "I'm kind of shy," 
she said, "and don't really like to ask for help, even when I need it, (especially from 
someone I don't know)." She reported her sister to be "the exact opposite," like her Mom. 
Jamie saw herself as not so "bad" now, particularly in smaller groups of people she 
knows. Giving presentations used to be hard but is doable now. 

Jamie: Yeah, I've grown a lot since then, believe it or not ... Like, I used to be 

worse. 
JK: Really, a lot? 
Jamie: Yeah. 

JK: Oh dear. You say, 'worse,' as if this is a bad thing. 
Jamie: Yeah.t 

JK: Like, if people say, 'This is bad, you need to speak more', or? 
Jamie: Well, like, whenever I had to do oral reports and stuff, it was very 

traumatic. 
JK: Oh, dear. 
Jamie: Whereas now I'd be able to get by. (Session 1) 



249 

In Session 3. 1 raised the question of wiiether Jamie felt I assumed that her quiet, 
non-participatory style was all "bad" as she labeled it. I asked her what she saw as 
advantages of her style and she immediately responded that she was "able to listen more. 
'Cause some people don't listen; they're just talking all the time." I experienced this as 
illustrative of how Jamie might experience me at times and I began to explain my own 
efforts as an outgoing extrovert with shy, introverted family members to modify my 
behavior and listen. I told Jamie (and reminded myself) how difficult I found it to listen 
to quiet people, to wait long enough for them to form their thoughts and answer; I was 
aware of how important in the recovery of her mathematics self it was for Jamie to fmd 
and express her voice and I had to allow that to happen. 

This led Jamie to discuss her mathematics ability in relation to family beliefs. She 
reported that her mother often said she passed her own "not good" mathematics genes to 
Jamie and her sister. "My dad, he's very good at math. My mom always said that 
unfortunately, me and my sister got our math genes from her 'cause she's not good" 
(Session 3). I questioned her mother's theory and reminded Jamie that we were gathering 
evidence that reflated that claim. 

Jamie used the words "good," "not bad," "bad," "not good" or "worse" to classify 
how she and her family did mathematics, to describe her progress in dealing with her 
shyness in school, and to describe her feelings. I wondered whether Jamie meant them as 
polarized judgments and if so how much they might be locking her into particular 
positions — if she (and the females in the family) was "not good" at mathematics or 
"do[es]n't do good at math," if storms were all "bad," if only others like her Dad were 



250 



"good" at mathematics, even contradictory evidence such as her Algebra I experience or 

her 95% could be discounted as anomalies. 

Jamie 's Attachment to Mathematics 

My first impression was that Jamie's mathematics cognition functioning level was 

very different from and considerably higher than Karen's. Her 95% grade and the story 

she told of how she achieved it spoke of a firm mathematical knowledge base, good 

trouble-shooting skills, and an ability to perform under pressure. Because we didn't have 

Jamie's Exam #1 with us,'"'" she had to recount her experience from memory. 

Jamie: Well, there was one part. . . I started doing it, and then I was like, 'Wait. 
That's not right!' So I went back and I changed it. Like, it was one of the 
ones that had to do with some of the earlier problems too. . . So I went back 
and I had to change everything, because I was getting all in my brain, like I 
was . . .how to do the wrong . . . wrong equation. Like I was doing the right 
one for a different one,'"'^ but not-- ... So I was getting them mixed up. ... 
But then I realized it, and I went back and fixed them all. ... 'Cause I was 
having problems, and I was, like, 'Why is this not coming out right?' ... 
And then I figured it out. 

Great! So did you feel good when you went back? And you were like, 
"Yeah!"? 

Yeah,t because I wasn't really sure at first; I was confused if it was right or 
not. ... It didn't really look — you know... So — but then after I fixed it, I 
was confident... 

And so what about it made you feel it was not right? 
Um, I think it was the answer I got. . . . Like, I think it was the 'z' score 
[standard normal deviate score — a transformed score indicating how many 
standard deviations a score is from the mean] or something. . . .And I got a 
really high number that was. . .not even on the chart. ... so I figured it was 
probably wrong, if it wasn't even on the chart. 
JK: Right. Cause zs only go up to, like 3 something- 

Jamie: Yeah, so then I went back and I was like, 'Oh no,' and I was all panicky, 

and then ... I realized what I did. So it was okay. . . I think I was doing the 
wrong thing for that [the sample standard deviation].... But then I noticed, 
so I fixed it. (Session 1) 



JK: 



Jamie: 



JK: 
Jamie: 



In this interchange, there was clear evidence of Jamie's robust number sense, her 
understanding of the statistical concepts, her use of letter symbols, her self-monitoring. 



251 

and her problem- solving strategies under stress. When we did look at her first test 
(Session 5 just before the optional comprehensive final), I saw what she had done (see 
Appendix L, Figures LI and L2). Jamie had remembered accurately. Jamie's number 
sense was illustrated by her realization that a z score that was too large was caused by a 
standard deviation 5 that was too small since the 5 is in the denominator of the z formula: 
Z = X - X . Jamie was clearly pleased with herself that she tracked down and 
corrected her error, especially because the realization of her error pushed her from her 
customary anxiety into a panic. As I listened to her I affirmed her masterflil handling of 
the situation. 

Unlike Karen's sense of "always" having struggled with mathematics, Jamie's 
variable history, including positives such as getting As through fourth grade and doing 
"good" in Algebra I, and her less categorical "usually" very tough pointed to the 
probability that her mathematics self had developed soundly-enough — arithmetically and 
algebraically — despite the storms. 

This evidence of good-enough mathematics iunctioning was tempered, however, 
by Jamie's high Abstraction Anxiety score'™' on the Feelings Survey pretest and her 
repeated declaration that "I don't do good at math." I wondered whether there could be a 
cognitive base for her high abstraction anxiety. In Session 2, 1 suggested Jamie complete 
the Algebra Test (see Appendix C) to see if her concept of variables was indeed related to 
her high abstraction anxiety. The results of the test showed that she was comfortably at 
Level 4, the highest level identified by the compilers of the test, but not at Level 5 the 
highest level postulated by Sokolowski (1997) whose adaptation of the test I used (p.97) 
(see Table 6.2 and Appendix L, Figure L3). Jamie thus began the course with an 



252 

understanding of the variable that I expected should be more than adequate for the 
task.'™" I interpreted the results of the Algebra Test to her: "You're a powerhouse, 
woman! This discounts my theory that your abstraction anxiety might be related to poor 
understanding of the variable. You're very sound right through level 4! Amazing! Not 
really amazing!. ..so it's just this issue, learning to ask for the help you need when you 
need it..." Jamie had more than a good-enough concept of the variable to negotiate this 
course successfully; she grinned. Perhaps her parent's expectations raised by Exam #1 
were not so unfounded! 

Jamie had some uncertainties about operations with the variable, but she was able 
to problem solve and check herself as she had on the first exam. Even on Test # 2, where 
she earned 74%, she had tried an inventive (though incorrect) strategy, on a problem 
dealing with the probability of success of .7 to solve a binomial probability question 
using the table that gave probabilities through only .5. 

It seemed that it was her stormy experiences with algebra, not an actual inability 
to do algebra that caused her abstraction anxiety to be so high. I hoped that these results 
might help allay her imcertainty about her mathematical ability, confirm that her algebra 
base was secure, alleviate much of her abstraction anxiety, and give her more confidence 
that she could do well in PSYC/STAT 104. 1 had also seen strong indications (e.g.. Exam 
#1, see Figures LI and L2) that Jamie's arithmetical understanding (including her 
operation and small number sense) were sound. This was confirmed when she took the 
Arithmetic for Statistics assessment with the posttests in class (see Table 6.1 below). I 
saw the main cognitive focus of our meetings then to be continued efforts to reconnect 



253 

her with her good-enough mathematics self, a process already begun with our Exam #1 

discussion and her Algebra Test resuhs. 

Emotional Conditions: Anxiety, Learned Helplessness, or Depression? 

Although her Testing anxiety score on the Mathematics Feelings survey was high, 
Jamie had successililly used mathematical trouble shooting in a crisis in Exam #1, even 
though the crisis put her into a state of panic. This was not the type of testing anxiety that 
interrupts or derails cognition. Rather, it seemed that it was a type of mathematics social 
anxiety confounded with mathematics and mathematics teacher separation anxiety (see 
chapter 3) that prevented Jamie from clarifying what she understood and from getting the 
help she needed, even when it was readily accessible. In class, during problem-working 
sessions, I had observed that Ann generally spent considerably less time with Jamie than 
with other students, although she checked over her shoulder almost as often as with 
others. During these sessions, Arm used a combination of roving checking over shoulders 
(and offering help if she saw trouble) and responding to cues from students: a raised 
hand, a head up as she went by, a verbal plea. Jamie gave such cues less frequently than 
other students. Ann's sense that Jamie was "fragile," seemed to inhibit her from offering 
Jamie more help (Interview 2). 

Jamie's beliefs about mathematics were slightly more procedural than conceptual 
on the continuum (a 2.7 on the 1 through 5 scale) and were more towards the 
toxic/negative rather than healthy/positive (a 2.5 on a 1 through 5 scale; a 3 is middle of 
the scale). On learned helplessness versus mastery orientation, Jamie had the most 
learned helpless score of the class (a 2 on a scale of 1 (learned helpless) through 5 
(mastery oriented) see Figure K4). However, her noticing and troubleshooting her error 



254 



on Exam #1, despite her panic, indicated a more mastery oriented than learned helpless 
approach in that situation. Again it seemed that it was not so much cognitive but a kind 
of social learned helplessness that was impeding her ability to take the initiative to get 
help she needed when she needed it. 

I wondered if Jamie's reported learned helplessness was indicative of the often 
linked situational depression but examination of Jamie's responses on the JMK 
Mathematics Affect Scales seemed to rule that out (see Appendix L, Figure L5). After the 
first session Jamie's responses had been largely positive. This was not surprising to me 
since she had just found out about her 95% on her first exam, although her responses 
during this second session while lower were still at or above the midpoint of the scale 
(even though she knew she had done worse on the second test). The only responses that 
were of some concern because of the level of negativity expressed (average of 46.5% 
positivity; five responses at or below the 50% mark) were Jamie's responses at Session 3 
by which time her expected low grade on Test #2 had been confirmed — a 74%. After that 
her responses bounced back and remained largely positive. Taking this positive affect 
with the strong indications that her anxiety was more central suggested that mathematics 
depression was not a real concern for Jamie. 
Identifying Jamie 's Central Relational Conflict 

My experience of Jamie's transference was that she saw me as no less dangerous 
than the teacher who had first sent her into hiding. If I had reacted to this transference as 
Aim did by staying away in order not to hurt Jamie, I would not have pursued Jamie to 
begin counseling. Her insights into her own shyness and introversion pointed to a 
conclusion that her central mathematics relational difficulty was multi-faceted anxiety 



255 

and should be the focus of the counseling. This anxiety not only increased in mathematics 
testing situations, it kept people at arm's length and stopped her from getting the help she 
needed and therefore frustrated her goal of "doing good" (but not too good) in 
mathematics. My interest was in the domains, triggers, and origins of the anxiety. Jamie 
spoke of her social anxiety and her success in overcoming it in public speaking. But this 
social anxiety influenced her behavior in other domains, specifically the mathematics 
classroom, and in relating to mathematics teachers. Her lack of interaction with anyone in 
class, effectively hiding while we faced each other around one rectangular table, her 
failure to make a follow-up appointment with me, her discomfort when I asked her a 
question in Study Group 1, and my worry (influenced by Jamie) about asking her a 
question in front of her peers, all spoke of her anxiety in relating to people, her wish to 
avoid them, and the demeanor that dissuaded them from interacting with her (at least in a 
mathematics setting). 

In the one-on-one setting I found Jamie more open and willing to connect with me 
than Karen had been, although at times, especially in the beginning, she exhibited 
discomfort (In my notes written immediately after Jamie's first session I wrote "At times 
Jamie seemed close to tears."). Jamie had done well in mathematics and related 
positively to her teachers through 4* grade, secure in a mathematical learning base. The 
mathematics separation anxiety that was connected to her 5* grade experience seemed to 
have been exacerbated by the subsequent experience of finding then losing a secure base 
in her Algebra I teacher, who she perceived as unavailable when she needed her later. 
This precalculus class experience also seemed to have made her attachment to the 
mathematics itself, particularly algebra, feel insecure. At this point Jamie did not know if 



256 

or how she might do well or poorly and she did not perceive mathematics teachers (or 
tutors) to be safe enough to ask for input and support. 

In addition to her mathematics social anxiety and her separation anxieties she 
seemed to have experienced a debilitating performance/fear-of-success anxiety on her 
second exam related to a combination of shyness and her family dynamics — her father's 
expectations and pressure versus her mother's acceptance of Jamie's lack of mathematics 
ability. 

The central conflict that was keeping her stuck seemed to be between her desire to 
succeed in the course, her uncertainty about her ability to succeed, and her sense that 
becoming conspicuous might endanger her in some way. She seemed trapped far from 
home, separated from mathematics teachers and her mathematics self and she saw no 
means to reconnect safely and inconspicuously. 
Central Counseling Focus 

I realized that if I stayed where Jamie's transference put me (leaving her alone so 
as not to endanger her) she would not become aware of her issues in a way that would 
allow her to change. In counseling therefore my focus was to reach past the protective 
shield Jamie had built up for herself, to disempower the objects of her anxiety. I had to 
provide myself as a secure mathematics teacher base, a smooth path with few surprises 
around the comers, and I had to help her find a way to assess the level of safety of the 
class and the instructor so that she could choose to access her as a safe support base. 
Ann's non- intrusive, respectful approach made the present class a good-enough secure 
base where this could happen for Jamie, if she were able to see it and was willing to take 
advantage of it. I also had to help Jamie recormect with her own sound mathematics 



257 



cognition so that she could proceed with her mathematics learning, secure in her 

mathematics (arithmetic and algebraic) base. 

Jamie's Course of Counseling: The Process of Brief Relational 

Mathematics Counseling, Session by Session 

(see Appendix L, Table LI for Jamie's schedule) 

Jamie 's Sessionl 

I found out a lot about Jamie's mathematics experiences and her current 
orientation at our first meeting. Because she had done so well on her first test and the 
next test was not imminent, there seemed to be the necessary leisure. Telling her story did 
not seem easy for Jamie — at times her color deepened and she seemed uncomfortable, 
sometimes close to tears — but that did not impede her or me from our exploration. My 
curiosity about her story, the connections, and the apparent contradictions seemed to help 
her become conscious of it in a piece for the first time — no one had asked about it before. 
Themes of Jamie's shyness and social anxieties evident in class and in study group, her 
variable success, the impact of teachers on her success and sense of mathematics self, and 
her personal preference for calm rather than storms — the absence of "up and down," — 
emerged. 

When we worked on linear regression problems I noticed that Jamie's arithmetic 
and use of algebra seemed adequate and she did not seem anxious as she worked. At the 
end of the session, as we worked some of the assigned homework problems in parallel, 
another student came for his appointment. Jamie finished the problem she was on and left 
without arranging our next meeting. 



258 

Jamie 's Session 2 

Again, as for the first meeting, although there was ample opportunity, Jamie did 
not approach me to make an appointment for a second meeting. I eventually approached 
her and we arranged to meet during the July 3 cancelled class time. Our second meeting 
took place during the sixth week of the course when she knew that she had not done as 
well on the second test but didn't know her grade. 

At this meeting we discussed a picture I thought seemed to be emerging. I showed 
Jamie her Survey Profile Summary (see Appendix L, Figure L4) where her testing anxiety 
and helplessness scores were the most extreme in the class and her abstraction anxiety 
high, exceeded only by that of another student. Discussion of learned helplessness (and 
its counterpart mastery orientation) and Jamie's extreme score led us away from the 
cognitive domain (Jamie's mastery recovery on Exam #1 seemed to discount helplessness 
in that domain. See p. 251 .) towards the relational domain. 

I asked if Jamie would have eventually approached me for an appointment if I 
hadn't approached her. She didn't know but from past experience it seemed unlikely, 
even though she had not done as well on Exam #2. I suggested that her learned 
helplessness might be more about this apparent inability to access help even when she 
knew she needed it — a type of relational helplessness — and that this was perhaps also 
illustrated by her behavior in class. When I asked Jamie why in class, for example, she 
didn't ask the questions she had in her mind, she replied, "That's my fear that I'll be 
wrong." When she was not convinced by my response: '"Asking a question, you don't 
have to be right!" I realized that to safely even ask a question, for her, it would have to be 
a "right" question, one that fit logically and unobtrusively into the context. Even during 



259 

problem-working sessions when students were working in pairs or alone and Jamie 

always alone, she would not ask Ann a question though Ann made herself available by 

circling the room and checking on each student's progress. Neither did she answer any 

questions during lecture-discussion sessions even when several students responded 

together and she knew the answer. The unconscious subtext seemed to be that she would 

draw attention to herself if the question or answer were "wrong." Why was I so 

concerned that Jamie speak in class? Perhaps it was just her style and of no consequence? 

I asked Jamie what she did if she did not understand what was going on or how to 

proceed and she replied, "Nothing." That concerned me and I believed it should have 

concerned her. We discussed a possible relationship between my asking her questions in 

study group in order to get her to verbalize and clarify her thinking (even when I knew it 

made her uncomfortable) and her exam results. 

I told Jamie that I would no longer approach her to make appointments. To do so 

would allow her to continue in her pattern of getting help only if she was required or if 

people like me pursued her (even though her "don't hurt me" demeanor made that 

unlikely). I wondered out loud if she might be able to practice getting the help she 

needed, and I suggested two homework assignments for her: (a) to set herself an 

assignment to ask or answer a question in class, (b) to approach me to make the 

appointment for our next meeting if she wanted one. I confessed to Jamie that I found it 

hard to let go of my practice of making appointments with her, and allow her to choose to 

do it or not: 

That's risky to me {giggle} 'cause . . .1 have the sense that you have so much 
potential and I have a sense that here are some of the clues as to why you don't do 
as well as you could and that's exciting to me {laugh}. I think, "We could really 



260 

get her over this hump," you know, doing math, so Td really, I'm like, '1 want to 
go get Jamie!" and that's my mother thing. (Session 2) 

When Jamie then took the Algebra Test to explore possible connections between 
Jamie's high Abstraction Anxiety score and her understanding of the algebraic variable, 
her sound level 4 pleased and surprised her and moved my "sense" of her mathematical 
potential to conviction of it. Jamie left after rehearsing her two relational assignments. 

During the next class after Session 2, Jamie asked Aim some questions during the 
problem- working session and she was pleased with herself At the end of the class, she 
came round the table, beaming, calendar in hand, to arrange an appointment with me. 
Jamie was making a move to throw off the hold of the bad internalized teacher presences 
from the past, overcome her social anxiety, and alleviate her separation anxiety, both in 
Arm's class and with me. 

Jamie 's Session 3 

Jamie wanted to go over her second test and Ann had given it to me so we could. 
After we had briefly discussed Jamie's significant achievement in asking Ann questions 
and making an appointment with me, that is much of what we spent the session on — 
analyzing her work in relation to her preparation and affect. This was the test for which 
she had been so anxious because of pressure she felt from her family to get the imlikely 
95% she had on the first test. Although her anxiety was elevated, Jamie again did not 
seem to have been cognitively derailed by it. Her difficulties, she realized lay mainly in 
insufficient preparation — "like I knew how to do the things but I didn't know what they 
were called" and she had only put formulas on her formula sheet, not what they referred 
to. That she lacked understanding of what it was she was (accurately) computing affected 
or was affected by her poor preparation for the letter symbol understanding section. We 



261 

had both made the same error in the multiple choice section and I shared with Jamie a 
significant computational error I had struggled with and corrected just in time. I was 
attempting to challenge the lines Jamie had drawn between "good" and "bad" at 
mathematics; I, like her, struggle and make errors. 

We began talking about Jamie's now feeling able to ask Ann some questions in 
class. She attributed some that to Arm: "That's partly because it was just her," which led 
to a discussion of how Jamie's "staying inside" behavior in mathematics class seemed to 
have begun with the distress of her 5* grade classroom experience and its effect on her 
shy personality. I suggested that as a 1 0-year-old, she had to survive what to her was a 
frightening situation, so she did what she could — "sit down and shut up so you don't 
bother her [the teacher]." But I wondered out loud with her whether, as a young adult, 
now Jamie might have more choices. Maybe she could now assess the safety of the 
classroom situation and decide whether she could participate. Jamie agreed that the small 
classroom and the positive, supportive emotional atmosphere Arm had created made the 
PSYC/STAT 104 classroom was such a situation and she had chosen to participate. 

Jamie had been filling out a JMK Mathematics Affect Scales at the end of each 
session (see Appendix L, Figure L5). At the end of Session 3, despite her having made 
such strides in personal interaction in relation to the class and getting help, her scores 
were the lowest yet, three of her seven responses (Items 2, 5, and 6) falling below the 
midpoint towards the negative end for the first time and another (Item 3) remaining there 
from before. Overall her responses seemed to indicate the presence of mild mathematics 
depression that had not been evident earlier. 



262 

Supervision 

On July 20, 1 met with Dr. P. for supervision. By then Jamie had taken Exam #3 
and earned an 84%, improving a whole letter grade over her Exam #2 grade (see 
Appendix L, Table L2). She and Karen were the only student participants whose grades 
did not drop by at least a half letter grade. I expressed my struggle as an out-going 
extrovert to be quiet and listen to Jamie, a shy introvert who preferred not to speak. Dr. P. 
encouraged me to invite Jamie to reflect on the changes she had made. "Commend her, 
give her a bouquet. Have her write a new metaphor." And he encouraged me to continue 
to my struggle to listen more and talk less, allowing, more, encouraging Jamie to express 
her voice (Dr. P., Supervision). 

Jamie 's Session 4 

Jamie again approached me for an appointment for a fourth session. And she did 
compose a new metaphor: 

JK: So your metaphor was a storm; what would it be now? 

Jamie: I kind of see it like it would be different for this class ...not necessarily 
math in general ...maybe partially sunny ... maybe bring an umbrella in 
case it turns to rain but it's okay to go outside, maybe, more, you know, 
because it's sunny I can go out in it, but I would still take my umbrella. 
(Session 4) 

This metaphor shows significant changes from Jamie's prior sense of 
endangerment in the mathematics class. Jamie's behavioral changes in the classroom 
situation — her little smiles, making eye contact, asking Ann questions in problem- 
working sessions, and her continuing to make appointments with me — were all outward 
indications of the changes she experienced. I observed myself doing better at waiting and 
listening for Jamie and she was now receiving Ann's offer of respect and safety. She felt 
safe to "go outside." She seemed to be resolving her conflict between fear of being 



263 

noticed and humiliated, and wanting to succeed in the course. She was finding that by 
going "outside" she had reconnected with mathematics teachers who were available and 
had not caused her damage, and had helped her reconnect with her mathematics self that 
she found to be good-enough for success in the course. 

Jamie's additional 6 points on the extra credit in-class power assignment brought 
her grade on Exam #3 to a 90%, an A". She reported that her father was very pleased. 
She did not show the increased anxiety she had in response to her high score on the first 
test, however. Jamie herself seemed encouraged and quietly determined, I surmise, 
because she was feeling more firmly attached to and was drawing on her own good- 
enough mathematics self Following Session 3, she had been more active and strategic in 
her preparation and found that she could change the outcome, so her father's expectations 
were no longer felt as external pressure to pull off another flukish feat, but rather were 
now more in line with her own realistic expectations, given what she now knew of her 
sound mathematical base and the importance and possibilities of strategic preparation. 

We worked together on questions Jamie might encounter on the Exam #4 
scheduled for that evening. When Jamie filled out the JMK Mathematic Affect Scales 
there was no longer any of the mild mathematics depression that she seemed to be 
experiencing at our last meeting (see Appendix L, Figure L5). 
Study Groups and the Final Exam (Exam #5) 

I did not meet again with Jamie one-to-one until after the final exam. She was at 
the study group just before Exam #4 with Ann and me, but she kept to herself; Jamie 
earned a 76%. Her symbol identification was perfect as for Exam #3 but her score on the 
multiple-choice had not improved irom the previous exam and this time she also lost 12 



264 

points on the computational section. Only one third of these errors were from lack of 
preparation or conceptual understanding of an analysis. So this score though similar to 
her score on Exam #2 meant something quite different about Jamie's grasp of the 
material and the process. 

Jamie came to the study group immediately before Exam #5. 1 asked individuals 
in turn to name the appropriate statistical test for scenarios I compiled from the text, and 
then we discussed the responses. Jamie responded incorrectly about a scenario requiring a 
two-way ANOVA when it was her turn but through discussion she understood the 
solution. Each student was involved in the others' questions. Jamie went on to earn a 
100% on Exam #5. 1 felt less anxious about causing Jamie trauma by askmg her questions 
at that final study group than I had at the first study group, and again the outcome was 
good. It gave her the opportunity to express and evaluate her thinking whereas merely 
thinking about it might have left misconceptions unchallenged. 

She had earned a 1 00% on her MINITAB presentation with Robin where she was 
poised and showed no signs of embarrassment. Thus, with her 100% on the fmal exam 
Jamie was getting a B^. With her father's encouragement she decided to take the 
comprehensive final to replace her lowest test score of 74%, hoping to bring her final 
grade up to an A". She asked me if we could meet once again to review all her exams as 
preparation. 

Jamie 's Session 5 

Session 5 was a marathon at a coffee shop on the Sunday evening before the 
comprehensive final exam. We reviewed each of her exams. Because the grade earned 
on each test did not necessarily reflect the level or quality of her mathematical thinking, I 



265 

decided it was important for Jamie's growing mathematics prowess to identify where she 
had thought well and to reduce the role of the grade as sole measure of her ability. It was 
also important, however, for Jamie to realize that issues other than mathematical 
understanding, such as clear communication and correct solutions, can be so important in 
real life application, that instructors use severe point penalties to emphasize this on an 
exam. Twice on Exam #4 Jamie had made the logical decision based on her (incorrect) 
calculations of statistics, but had 4 points deducted on each because these were incorrect 
decisions for the problem. Arm had also deducted points for Jamie's technically accurate 
but poorly communicated defmitions of symbols. Jamie was able to see that her grade on 
Exam #4 undervalued her actual mathematical thinking and ability; I nevertheless 
emphasized that the grade J/J enforce the importance of her improving accuracy and 
clarity for her chosen field of psychology. 

We also discussed changes in Jamie's responses to the Mathematics Feelings and 
Mathematics Beliefs surveys. She had made substantial changes on each of the anxiety 
scales. By the end of the course her score on testing anxiety had gone down from high to 
moderate (a 17% decrease,'""" see Jamie's post-scores on Figure L2, Appendix L). My 
anxiety about Jamie and her mathematical learning and my inclination to control and 
mother had also decreased as she took more control and internalized realistic expectations 
of herself. Jamie's beliefs on the Learned Helpless versus Mastery Oriented scale had 
changed very little but her social learned helplessness, at least in this setting, had abated 
considerably. Jamie earned a 71% on the comprehensive final exam (80% on the 
computational part), and could not replace any of her earUer test grades, so she ended the 
class with a B^. 



266 

Jamie 's and My Final Evaluations 
On Jamie's One-On-One Mathematics Counseling Evaluation she described her 
initial motivation for signing up to meet with me was "so that I could get a better grade in 
the course." Since I only learned after the course from Jamie that she had been repeating 
PSYC/STAT 104, 1 speculate that initially she had not disclosed this because of her "fear 
that I'll be wrong" and thus conspicuous and censured by a mathematics teacher whom 
she did not know (me) and who was not to be trusted to do anything but humiliate and 
abandon her, as past teachers had done. Her end-of-course written comment about "a 
better grade" may have been an indication of her now feeling safe to let me know, 
perhaps also affirming her trust that I could know that she had not done well in the course 
before without rejecting her. 

Asked whether her motivation had changed during the course, Jamie indicated 
that she recognized the focus was primarily relational, "Kind of, I realized it was more 
about my feelings and confidence in my math ability, th[a]n any real problems with the 
math course work" {One-on-One Counseling Evaluation, archived). 

Jamie had learned how to ask for help in this course but it seemed this experience 
was not enough for her to do it in a new class. Unless the class was structured like her 
English composition class with required meetings with peers and instructor, or had a 
resident class-link tutor who initiated the contact, I concluded Jamie would probably 
continue to be an involuntary loner. Although it had for this class, for other mathematics 
classes, her conflict between wanting to do well (and knowing that likely means getting 
help from and working with strangers) and fear of being conspicuous had not been 
resolved and the latter would probably predominate. She /za<i however, become aware 



267 

through counseUng that she could judge the safety of the situation and the instructors and 
helpers and not feel compelled to hide no matter what. 

Jamie had done well in the PS YC/STAT 1 04 by reattaching to her sound-enough 
mathematics self and to safe mathematics teachers/classroom. I had learned to wait, 
listen, and affirm her strengths as well as challenge her to confront her fears. I (and Ann, 
once I had helped Jamie see) had provided her with good-enough objects to replace the 
bad 5"" grade teacher internalized object (presence). I had become a secure enough 
base — a smooth level path, with a gradual incline — from which she could experience this 
class, not as a storm any longer but now as a "partially sunny [day]" where she could go 
"outside." It is not possible to say what grade Jamie would have earned in PSYC/STAT 
104 without counseling, but she almost certainly would have remained hidden, the 
instructor would have tiptoed around her, she would have remained isolated from her 
peers except for the required contact over the computer module presentation, and her 
questions and comments would have remained unspoken. Most importantly Jamie's sense 
of her mathematics self would likely not have changed. If I had not examined her 
transference and my countertransference reactions I might not have pursued counseling 
with Jamie at all. If I had only gone by Jamie's responses on my traditional anxiety and 
belief surveys and not delved with her for the underlying meanings they signaled, if I had 
not explored her metaphors and tracked her progress with the JMK Scales, in other words 
if I had taken a traditional approach instead of brief relational mathematics counseling 
with Jamie, she is likely to have remained an under-confident involuntary loner achieving 
variable results over which she felt little control — always afraid of the storms of 
incomprehension, anxiety, and unwanted attention. 



268 

Epilogue 

Jamie decided not to try Finite Math again to flilfill her quantitative reasoning 
general education requirement, but to take a logic course instead. She has not let me 
know how it went. Jamie now knows she could assess the relational safety of the 
instructor and the class to see if she might go "outside," ask questions, and ask for help, 
and she has a budding understanding that she was in fact quite capable of doing 
mathematics. If Jamie perceives a new situation as benign enough so that she does not 
regress and go back into hiding with "you are dangerous to me; don't come near me" 
transference, an attentive instructor or class- link tutor might feel less reluctant to 
approach her to offer help and she is more likely to accept such offers. If the mathematics 
counselor or tutor waits for Jamie to come to the Learning Assistance Center or make 
contact with the instructor or even the class link tutor, it is likely they will wait in vain 
and Jamie will not receive the help she needs. 

MULDER'S COURSE OF COUNSELING 

Mulder™"" exuded an outgoing social energy. During the first class lecture 
discussion on the scientific method, he was actively involved™", telling classmates of his 
research project on centipedes' attacking postures. During problem-working sessions he 
always worked with any neighbor willing to engage. Ann thought he seemed "smart" and 
"on the ball [with respect to] his research experience into caterpillar aggression" — but not 
likely to put in the effort needed to succeed in the class and not very committed 
(Interview 2, Class 1). 

When he foimd out that I was available as a tutor, Mulder was enthusiastic. He 
didn't think he would need much help with statistics (He had used some statistics for his 



269 

biology research projects.), but he thought he might for his finite math class — that 
seemed more challenging to him. So Mulder signed up for mathematics counseling once 
a week. He struck me as a charming scallywag. In fact, I called him that once. He 
seemed busy, mischievous, stubborn, and somewhat of an opportunist but he was 
confident that he could handle PS YC/STAT 1 04 fine, so I believed we would focus 
mostly on finite math. 

However, Mulder and I soon found reason to suspect that his confidence was 
perhaps overconfidence. He earned a 'D"' on his first exam and thus began a quest unlike 
any either of us had been on before. I found I could deal with anxious and underconfident 
students like Jamie, using relational approaches to get at the roots of her anxieties; I could 
even overcome the despair that depressed and underconfident students like Karen threw 
me into because both of these students knew that they needed help. But how was I to use 
relational approaches to recognize that a student with a social, confident, and up-beat 
demeanor might actually be overconfident and that he might then be drawing me into 
believing he was less needy than he really was? Then once I recognized this, how could I 
help when he seemed to have all the answers? And for Mulder this seemed to be new too. 
It turned out that he had never really tested his "I know I can do math" theory by actually 
trying to do it well and he had not worked with someone who was trying to support him 
in that endeavor. 

Sometimes Mulder developed what appeared to be indirect and to me illogical 
schemes to improve his achievement; at other times he stubbornly resisted the 
mathematics he found did not yield to these devices. He did improve his computational 
grade and then his grade on symbol identification improved, but on exam after exam he 



270 

failed to improve his conceptual section score. The narrative that follows chronicles how 
we struggled and how relational counseling insights and approaches I used not only 
helped Mulder resolve the conflicts that hindered his success but also helped me grow as 
a mathematics counselor. 

Mulder was a 20 year-old white man who was a biology major at a small 
university in the Midwest. He was home for the summer, taking Finite Mathematics — 
MATH 120— in addition to PSYC/STAT104 at Brookwood State. The fmite 
mathematics course was required for his major but not statistics. He had the option of 
transferring his statistics credits for elective credit if he did well enough. 

Mulder was an only child. He was short, muscular, and fit — ^participating in both 
soccer and track (Class 1 ). The last mathematics course he took was Algebra II as a 
junior in high school; he reported that he earned Cs in mathematics classes then. He 
mdicated that he hoped for a B and expected a B in the PSYC/STAT course (Pre-Test 
Mathematics History Survey), both overestimates perhaps, given his history. 
Mulder 's Metaphor: Fox Mulder Searching for Aliens 

Mulder asked if he could think about choosing a metaphor "because I really — I 
don't know that I could say for a while" so I suggested we come back to it later. When 
we did come back to it I asked him if he would rather do a drawing of himself doing 
mathematics, but a metaphor came to him, "For me math is like Mulder searchmg for 
aliens. I am searching why I make math so difficult for myself" He referred to Fox 
Mulder from The X-Files,™^ a popular science fiction television program. Mulder 
explained further, "I have confidence in everything else I do. It's not that I don't have 
confidence [in my ability to do mathematics], but it's just like — ^I know what I'm doing. 



271 

but I can't explain it to other people." I wondered how this related to his "mak[ing] math 

so difficult" for himself And what if anything did the metaphor tell us about how Mulder 

saw mathematics? 

Student-Mathematics Relationships: Mathematics as a Search for Aliens 

When I asked Mulder about how mathematics had been for him, he responded 

"It's never been my favorite." Later, "It's, like, it's the only thing that ever gives me any 

problems." It was from freshman year of high school that mathematics seemed to have 

become an issue for him. His theory was that it was his lack of effort rather than low 

ability that accounted for his difficulties, yet he had not tested his theory by putting in 

that effort even after he "realized" that was his problem. He avoided mathematics 

altogether his senior year because he knew he would not do well in it: He wanted to "save 

[his] grade point average". 

I formed this attitude in high school, you know, high school, if I had really, really 
tried in high school, I could have done really, really well [in mathematics]. It 
wasn't 'till the end of my freshman year I realized and I still don't think I try as 
much as I should, you know ... It's just a matter of applying it and taking 
advantage of it ... study skills in high school weren't that great [I did] three 
sports a year. I did real well my senior year because I took no math ...I was 
enrolled in trigonometry and precalc but 1 dropped it because I wanted to save my 
grade point average. (Session 3) 

Mulder had done better in Geometry than in Algebra II even though he had 

"thoroughly slept through it" and he put that down to the difference between the teachers 

rather than differential ability or a preference for that type of mathematics. 

I got 'C's in all these [mathematics] classes. This one [pointing to Geometry on 
the list] I thoroughly slept through; I'm not lying... ironically enough I did better in 
this class [Geometry] than in this one [Algebra II]. It was the teachers. (Session 

3) 



272 

He could attribute his low performance to not trying or to sleeping, so his belief in 
his potential ability to do it was preserved especially since "my uncle is getting his 
masters, great student and my dad's really smart so it's kind of like a thing I know I 
have" (Session 3). 

I found my initial reactions to Mulder and his prospects in the class were quite 
different from my reactions to Karen, even though her grade on the first test was almost 
the same (62% compared with Mulder's 63%). His metaphor was active, if somewhat 
self-defeating, and he seemed willing to engage. Mulder was positive about his 
mathematical potential. I was drawn into his confidence and considered then neither that 
his knowledge base might be weak nor his underlying mathematics self-esteem low. 
Because he had earned Cs through Algebra II with lots of sleeping, and not really trying, 
the result may have been a relatively underdeveloped mathematics self I eventually 
found considerable evidence to support this conjecture. 

After the course it was confirmed that Mulder did have a minimal algebra 
background for college though this was not obvious to me during the course. When he 
took the Algebra Test after the conclusion of the course, he tested at a low level 2, 
indicating that he, like Karen, had not yet developed an understanding of letter symbols at 
least as specific unknowns or generalized numbers (and in some cases as true variables) 
nor could he coordinate operations using them (see Table 6.2). That perhaps explains 
why his formula sheet for the first test had been so inadequate for his needs: he, like 
Karen, needed detailed formula sheets for exams that interpreted formulae into columns 
and step by step procedures. Unlike Karen though, he seemed relatively arithmetically 



273 

sound, with accuracy and confidence in his number and operation sense (see Session 1 
and Table 6.1).'°°^ 

Initially I was taken in by Mulder's sound-enough arithmetic and confident take- 
charge approach. I did not become conscious of his real deficits with respect to the 
algebraic variable and related concepts until later in the course. I now see that Mulder's 
low understanding of the algebraic variable, his more procedural than conceptual beliefs 
about mathematics (2.9 on a scale of 1 through 5 on Beliefs Survey, see Appendix M, 
Figure Ml), and his poor high school preparation seemed to have combined to make the 
statistics almost alien to him, especially the conceptual aspects that required him to 
understand and communicate in earthly rather than alien terms. Unconsciously at least, 
these factors were almost certainly calling into question for him his own ability. Maybe it 
was not just about effort. Maybe he really could not do it. 
Student-Teacher-Self Relationships: 

Mulder complained about Ann's lecturing style (She "jumps around a lot.") Later 

in that first session he said of Aim's lecturing: "It's not that she goes through it so fast; 

it's just I have a hard time following her" (Session 1). His references to past teachers 

were in a similar vein. He attributed doing better in Geometry that Algebra to his 

teacher's different approaches. 

Mulder: It was the teachers. 

JK: You seem to react to teachers 

Mulder: Yeah, I do 

JK: And it seems to affect how you do in class? 

Mulder: Yeah. (Session 3) 

Present struggles with teachers seemed to be closely Imked to Mulder's struggles 

to understand the course content and gave me clues to the nature of his past struggles. He 



274 

strongly preferred his finite teacher's direct approach, "[S]he tells us how to do it and she 
tells us why and how to use it" (Session 3). This teacher made explicit the links among 
concepts, procedures, and applications and she demonstrated the procedures. By contrast 
Ann had students work out how to do the problems for themselves during problem- 
working sessions, after she presented concepts involved. It seemed hard for Mulder to see 
how the concepts discussed in the lecture related to the problems worked later, even that 
they were related. And the struggle seemed to be exacerbated by Mulder's auditory 
processing difficulties and his compensatory visual memorization strategies. 

Learning/processing style and Mulder-teacher relationships. A pattern of 
Mulder's difficulties in understanding and expressing new knowledge through his 
auditory and verbal channels emerged. This was evidenced in his relative difficulty with 
finding a metaphor, following Ann's lectures, and in his linking his "Mulder" metaphor 
to difficulties he made for himself in mathematics, especially in explaining what he 
understood. Initially my realization of this difficulty was masked by Mulder's outgoing 
social learning style and I speculated that other mathematics teachers may have been 
similarly misled. I began to wonder whether Mulder was making it hard for himself or if 
it was a processing difficulty that he blamed on himself Perhaps it was a combination of 
factors. Perhaps what he labeled as his laziness was, in part, avoidance of these primarily 
verbal study tasks he found difficult. 

The seemingly illogical visual memorization strategies I observed him using 
perhaps served to compensate for his auditory struggles, and his perception was that he 
did better on assessments that required visual recognition of material. For example, as 
Mulder tried to understand why he did not do as well as he expected on certain exams he 



275 

cited testing anxiety as a factor for all exams except "practicals," (those requiring 
identification of visually discriminated materials: "Like, a bone practical [where he had 
to identify and describe the function of bones of humans or other animals]" He always 
got As on those [Session 3].) Did Mulder's have a global, visual-pictorial, mathematics 
learning style II (Davidson, 1983; Krutetskii, 1976)? Or was his approach the result of 
continued use of strategies he had developed to compensate for auditory/verbal 
processing difficulties? Or some combination?™"' It seemed that he was not easily 
classifiable but I began to wonder if his atypical approaches to mathematics learning 
might not only have negatively affected his level of mathematics understanding, but also 
how he was perceived by his teachers. 

I needed to explore with him what effect these approaches had on his mathematics 
self development. I needed to know how teachers had reacted to him and what effect that 
had on him. I myself reacted with amazement and sometimes horror to what seemed to 
me to be a lack of observable logic in some of his tactics (see Sessions 4 and 5). Mulder's 
approach seemed consistent with his metaphor at least; he did indeed seem to be using 
alien methods to search for his aliens. 

But these methods looked enough like attempts to avoid hard work that I 
speculated that his mathematics teachers had not only perceived him as capable (because 
of his confident upbeat demeanor) but lazy, but also labeled him thus. Indeed this was 
how Ann saw him: smart but not likely to put in the effort. Mulder's constant concern 
that he might be perceived to be lazy ("I hate doing this ... It's just — it makes me feel 
lazy" when admitting to putting work off 'a lot' when he filled in the JMK Affect Scales, 
Session 1) and his repeated description of himself as lazy about doing mathematics 



276 

supported this conjecture that he was used to being labeled thus and had taken it on 

himself. Perhaps the "capable but lazy" label had become a shield for his possibly 

incapable self and, if so, it may also have functioned as a trap, hindering him from doing 

what he needed to do to develop his capabilities and deterring teachers, whose help he 

needed, from helping him. 

Emotional Conditions: Anxiety, Learned Helplessness, Depression, 
or Grandiosity? 

Anxiety. Mulder certainly didn't strike me as anxious. But after his poor showing 
on Exam #1, he brought up testing anxiety as one of the factors he believed was operating 
against him, especially on tests like mathematics tests that were not visual memory 
oriented "practicals." I had not highlighted any of his average anxiety scores on the 
Feelings Survey for discussion with him (see Appendix M, Figure Ml) because each fell 
in the middle of the class range and was not extreme compared with the class. However, 
his testing anxiety averaged at a little above moderate (3.1 on the 1 through 5 scale) and 
could be considered high for a physical science-oriented student and even for a social 
science student if compared with means Suinn (1972) reported on the Mathematics 
Anxiety Rating Scale (MARS). '""'" 

Was Mulder's anxiety a normal reaction to a challenge he was not adequately 
prepared for or something more than that? It seemed feasible that it was linked with his 
history of not having done well on mathematics exams and an underlying belief that he 
may not be able to do it. That combined with lack of strategic preparation for this exam 
to compensate for his mathematics deficits (e.g., well constructed formula sheet, strategic 
practice of target problems) would give good reason for considerable but normal anxiety. 



277 

Depression. And Mulder gave me the impression of being anything but negative 
or depressed. This observation seemed to be confirmed when he completed the JMK 
Affect Scales during his first session. Apart from his extreme negative response (a lot) to 
putting work off" all other responses were at or above the mid point (see Appendix M, 
Figure M2, responses marked 1). His average positivity on the scales was approximately 
55% or 64% if the "putting off work" item were removed (see Appendix M, Table M3). 
Given Mulder's poor showing on Exam #1, rather than indicating mathematics 
depression, his responses perhaps showed the opposite, mathematics optimism.™"^ . 

Learned helplessness. Although depression was not an issue for Mulder, learned 
helplessness did seem to be. When I showed him his low learned helpless score average, 
Mulder responded, "I think it's math. Any other thing I'd be up here [pointing to the 
Mastery Oriented end of the scale]" (see Appendix M, Figure Ml). Perhaps this was a 
chink in his up beat armor that I initially did not explore. Although Mulder saw 
mathematics as somewhat more procedural than conceptual and his approaches seemed 
procedural, his achievement motivation became more for learning than for performance 
over the summer (Items 4, 7, 9, and 10, Part I, Beliefs Survey and Appendix H, Table H3). 
Perhaps like Karen, he wanted to understand the material but used procedural approaches 
both by habit and also because he did not feel capable of achieving that understanding. 

Grandiosity. Mulder's emotional response to his mathematics challenges did not 
seem to be marked by anxiety or depression that could be considered unrealistic. The 
evidence seemed to be pointing to grandiosity. It seemed that Mulder might have 
developed an overconfident demeanor combined with relative lack of effort and indirect 
approaches in order to protect an underdeveloped mathematics self that was 



278 

compromised by his learning style challenges: "I know I can do really, really well ... 

[but] I don't really try/I thoroughly sleep [through class]." 

Identifying Mulder 's Central Relational Conflict 

Mulder now faced a dilemma. He wanted a B in the two summer mathematics 

classes he was taking to make it worth transferring the grades. His high school tactics 

would not work but if he actually tried he risked being found out. On the other hand he 

did not want to be considered lazy. When I asked him about how much work he did for a 

finite exam, at first he denied doing any work. This seemed to express his grandiose 

stance (I can do well; I don't need to work at it.). But then he conceded that he had 

practiced but only some of the problems (perhaps his "I don't want you to think I'm lazy" 

stance) 

Mulder: And I don't — ^I've never really sat down and done practice problems 
before a test. 

JK: And you didn't do practice problems? 

Mulder: No. 

JK: Was it stuff you were already familiar with? 

Mulder: No. 

JK: No? But you got it Irom the class? 

Mulder: Yeah. I knew how to do it. I did — I did some of the homework; I don't 

do all of the homework but I do some of it. 

JK: Just pick a few things? 

Mulder: That I need — that I need to work on. (Session 1) 

Confounding his difficukies were the very real challenges that his auditory 

processing difficulties, his compensatory visual strategies, and his poor mathematics 

preparation posed, especially as Mulder did not seem to be aware of them or their 

potential for sabotaging his success. 



279 

Mulder and Me Dealing Jointly with Aliens: 
Relational Counseling for Mulder 

The Focus of Relational Counseling 

I realized that relational counseling should focus on helping Mulder and me 
become aware of the conflict between his competing goals — to do well in the course but 
also to protect his underdeveloped self. Perhaps the very defenses he was using to protect 
his mathematics self were what were "mak[ing] mathematics so difficult for [himjself " I 
would have to recognize that Mulder's grandiosity might be masking an underdeveloped 
and vulnerable mathematics self. He was so convincing and I found myself believing his 
grandiose view and not attending to his real challenges. I was likely falling into a pattern 
of former teachers — believing him, being disappointed, getting frustrated, scolding and 
pushing hum, and even giving up — and not offering him what he really needed. Not only 
would Mulder need to become aware and change, but I would also have to change my 
approach in order for him to feel safe enough to drop his counterproductive defenses. 
And he might need me to change before he could. I realized that we were unlikely to 
resolve his conflict unless I could work out how I should change. 
The Focus of Mathematics Counseling 

Because my preferred learning style is strongly auditory, I had to be aware of the 
risks of devaluing Mulder's mathematical learning approach, simply because it was 
different from mine. Instead I needed to accept and try to understand how his visual- 
memorization, his procedural mastery, and his social style both facilitated and impeded 
his mathematics learning. How could I help him use his strengths and preferences to help 
rather than stand in the way of his grasp of the mathematics? It became clear to me that 



280 

the strategic mathematics tutoring focus should be on Mulder's finding a way of seeing 
this alien mathematics in a more accessible, logical, earthly form.'™" 

Mulder's Course of Counseling Session by Session 
(see Appendix M, Table Ml for Mulder's schedule) 

Mulder 's Session 1 

Mulder was doing well in his finite math course, but he had done poorly on the 
first PSYC/STAT exam (63%). He had failed the multiple-choice conceptual section, 
with the lowest score in the class (see chapter 5, Figure 5.2). At times he had failed to 
follow directions ^°°"' and at others he did not know the information adequately so he 
guessed rather than trying to work them out from the context. His computational score 
was less extreme but still only 65% correct. He had lost only one point on the symbol 
identification part, but he had only named the symbols and not defined them as was 
required. Ann said she had been lenient in her grading on this section because it was the 
first exam. 

When we examined how Mulder had prepared for the first test where he had done 
so poorly he identified the fact that his formula sheet was not adequate and he had no 
direct information (e.g., a quiz) to guide him to work out how Ann tested so he had not 
prepared sufficiently or strategically enough. These factors seemed to give good reason 
for the testing anxiety he said he had suffered. 

Although he had not done well on the computational part of the exam,'™'"" Mulder 
seemed to have solid number sense and no problems with decimals so I was not alarmed 
as I was for Karen about his prospects. The questions involving number or decimal sense 



281 

(e.g., deciding on real limits for the weight of 0.35 grams of cheese) were answered 
correctly. 

As we discussed how to prepare for Exam #2, Mulder seemed to be astutely 
assessing the mathematical tasks required for him in the computational section. "They're 
not really word problems, you know. The information's there and the equation's there 
and she shows us how to set everything up, and I understand all that." Although perhaps 
globally positive rather than realistic, he did seem to have pinpointed a crucial problem 
with his first exam, that is, he had not set up his formula sheet adequately. If he had, "It's 
easy to write down the equation, say what the ground rules are, and then plug the 
numbers in." He saw the mathematical tasks as procedural and felt that he could manage 
that. Mulder seemed confident that he could remedy the situation in Exam #2 by 
improving his formula sheet and studying the procedures now he knew how Aim tested. 

Mulder seemed to consider the computational and conceptual sections of the 
exams as separate, requiring different types of preparation and despite his low score on 
the conceptual he commented, "I did all right on the conceptual part" and for the next test 
it was the computational part he was going to focus on. Later in the session Mulder did 
concede though, "Obviously, I need to spend more time on the conceptual." 

Although Mulder's decision on how to improve his computational preparation did 
not include understanding and linking the concepts, I did not pursue it, thinking maybe 
his plan would work. I was concerned about his conceptual understanding of symbols (on 
Exam #2, Ann would require that) but when I made a suggestion, he was defensive and 
claimed he had already done what I suggested. I also suggested ways of tackling his 
multiple-choice challenges but felt some resistance to my reconmiendations. 



282 

When we turned to working on the statistical procedures to be tested on Exam #2 
using both notes and diagrams, Mulder had some questions about the statistics but 
seemed to have control of straight line equations needed for regression. 

It was in this first session that I became aware that Mulder might have verbal and 
auditory processing problems. He talked about his struggles with following Ann's 
lecturing approach, he had more trouble than any other participant in composing his 
metaphor, and he had failed the conceptual portion on Exam #1.1 decided that at our next 
session I should try to help Mulder work out ways to compensate for his processing 
difficulties so he could make the conceptual connections with the procedures that he 
seemed to think he was capable of mastering. At this point in our relationship I thought 
that this would simply involve beginning with his procedural competence and working in 
parallel as I modeled finding conceptual connections. 

Mulder's Session 2 and Session 3: The Conflict Emerges 

Mulder 's Session 2 
Although the focus of this session was on exam preparation for a fmite 
mathematics exam in an hour,™"'" Mulder also tried to make sense of what was 
happening in PSYC/STAT 104 in terms of how he had previously done in mathematics 
courses. He had taken Exam #2 the night before and he knew that he had done badly 
again on the conceptual part of Exam #2. He had lost 19 points out of 50, including 5 of 
the 6 points for symbol identification — a D; Aim had shown him his score when he 
handed in his computational section. However, he was fairly certain that he had the 
computation 100% correct. Mulder had conflicting theories about whether historically he 
was better at the "math part" or the "conceptual." Now he was irritated by the fact that he 



283 

had done ' A^' work on what he considered the mathematics but was being denied 
recognition for that because of this "other stuff." 

I shared my ideas about what might be happening and proposed that he had a 
theory about what was and what was not mathematics and that, according to that theory, 
the conceptual part of PSYC/STAT was not mathematics. In addition, I told him that I 
saw him as a strong-minded person who acted on his theories, and in this case he was 
rejecting the conceptual aspects and concentrating on what he saw as real mathematics — 
the computation. Mulder agreed that I was accurate but explained, "I've always thought 
math was the harder part for me so that's what I've been concentrating on in the lecture. 
Everything seems to be centered around that formula, so I concentrate on that formula, on 
how to do that formula rather than taking it all in." With his poor result on the 
computational part of Exam #1, concentrating on that formula was indeed an important 
element of his recovery strategy. After all, it had led to success on the computation 
section of Exam #2. 1 was concerned about his rejection of "taking it all in" because it 
seemed tantamount to his deciding to dismiss the conceptual aspects of the course and not 
make the conceptual link to the computational. Did Mulder think that he was not able to 
do both or that he should not have to do both? It seemed that unconsciously he felt he was 
not able; consciously he insisted he should not have to. 

We were at this point still living in the initial transference-countertransference 
relationship. I had assumed Mulder's transference of past teachers so I had higher 
expectations of him that were reasonable and 1 was becoming frustrated when he would 
not or could not deliver on his confident plans. My countertransference reaction was to 



284 

push him, to accuse him of avoiding tasks (e.g., the conceptual), and to nag him with 
direct advice, expecting that he could get his act together. 

Mulder 's Session 3 
At this point Mulder was frustrated, "I don't know what to do! ... I don't know 
what to expect on the multiple-choice. ...We've had two exams now and I can't work out 
what it'll be." I also felt at a bit of a loss. I recalled from Session 1 that Mulder was 
ambivalent about doing the homework problems from the text because he was confident 
about the computational part of the exam, so I asked if he had done the homework this 
time. I also asked about the first text problems from each chapter set that were 
conceptual questions like: "What is the range of values that a correlation coefficient may 
take?" and "From each scatter plot in. the accompanying figure (parts a-f, on page 124) 
determine whether the relationship is ...positive of negative ...perfect or imperfect" 
(Pagano, p. 123). Mulder replied somewhat indignantly, "Those are the ones I did ... and 
I wrote them down [the answers]." But when we looked at his conceptual Exam #2 
errors, though, it seemed that he had not linked the concepts from this homework to their 
numerical meaning. For example, he had responded with 0.75 as the correlation 
coefficient that indicated the greatest strength on one question where the correct response 
was -0.80 because its magnitude is greater. Mulder insisted that he had read his class 
notes and the book a few times to prepare for the conceptual section. I suggested that 
only reading was probably too passive to be helpful. Further I suggested that he might 
even be he was even stopping himself from really learning it in some ways, resisting it 
because he did not beheve he should have to learn it. His sheepish reaction seemed to 
confirm my supposition. 



285 

As we continued to discuss what might be hindering Mulder's success I brought 
up his low learned helpless score on the Beliefs survey. When he said it was only with 
math that he was like that, I suggested that he could change it even in mathematics: He 
seemed to be doing well in his finite class, and could do so even in this statistics course. 
Mulder seemed skeptical until he remembered that he had recently experienced not 
giving up on a math problem. On the computational part of Exam #2 he had initially 
made an error that led to what seemed to hrni to be anomalous results. "I sat there a long 
time [looking at it] and I realized my standard error was wrong. It was way too big." He 
went back to find and correct the initial error in his calculations and then to fix all the 
computations affected: a mastery oriented response he now recognized. But could he do 
that with the conceptual section on the test? 

I recommended that he use study guide multiple-choice questions to prepare 
better for the conceptual section. That we focused on the contents of the previous test 
rather than on material for the upcoming test was not strategic but tackling these 
multiple-choice questions highlighted Mulder's misconceptions about material that would 
continue to be needed"™" and his ineffective study methods, especially on symbol 
defmitions and their links to the calculations that would be on the next test. Mulder 
agreed that this new tack of working on multiple-choice questions should help. I gave 
him copies of sets of multiple-choice practice questions for each chapter to be covered on 
the next exam for homework. He emphasized as he left, however, that at our next session 
just before Exam #3 we should review all the symbols because he had done so badly on 
the symbol section of Exam #2. 

Mulder 's Session 4: The Central Relational Conflict Becomes Clearer 



286 

Session 4 took place on the day of Exam #3. Mulder was tired, grumpy, and 
oppositional. I asked about what he had done to prepare for the exam. I had prepared 
practice materials using the problems we had done in class but I had removed any 
reference to the type of statistical test required to solve them so students could practice 
also identifying the test. I had also prepared an empty flow chart template for the 
statistical tests that would be on the exam for students to fill in as their formula sheet if 
they desired. Mulder was taken aback that he might be required to identify which 
statistical test was appropriate because he had been certain Ann had said that she would 
tell us what statistical test to use with each problem and we would not be expected to 
identify what test was necessary until chapter 19 and Exam #5. I told him that my e-mail 
exchange with Aim on the subject left the question open. He was not happy, grumbling 
about curve balls. 

Mulder grabbed one of the problem sheets, declaring that by looking at a question 
he could identify the statistical test required. Rather than analyzing the problem 
statement, he tried to remember by the look of the problem and the order it had been 
presented in class, but he remembered incorrectly. Another strategy was to identify the 
type by whether the data were presented in columns or as an already computed statistic. 
Again, he was incorrect. He made little attempt to read the questions and understand the 
situation or experimental design. I remonstrated and insisted that as a "bright man" he 
could and should think about the questions. 

Mulder ducked my comments and moved to a discussion of symbols. I set up a 
divided page to sort the population symbols from sample symbols and we began to 
discuss how to make decisions on tests. The single sample tests went smoothly but when 



287 

we returned to the two sample tests Mulder again tried to use his memory of the class 
when students worked them. He seemed to enjoy my frustration with his approach. I 
realized Mulder had not understood a central concept — ^that the words "independent" and 
"correlated" described the groups or samples not the data numbers. However, he insisted 
that to do a problem correctly on the exam he did not need to understand such 
distinctions, saying, "I would have figured it out because I would have looked at my 
equations and I would have figured out what went where." He then predicted that Ann 
would give the alternative type (SS| and SS2) on the exam because one type of already 
computed statistic (s,^ and Sj^) had already been given on a class problem. This 
speculation seemed illogical and risky to me. I was quite alarmed by how Mulder was 
orienting himself to the exam and he seemed to be enjoying my alarm. 

Next we engaged in a much-needed discussion of symbols; I drew a reluctant 
Mulder into making links with symbols he already knew. We discussed how to identify 
sample mean symbols'^ and looked at what might be a logical value for the population 
mean of difference scores, [Id ; because of the null concept of no difference in the 
problems the class was mastering, |a,D should be zero in the null hypothesis statement and 
therefore in the formula. Mulder ultimately got that wrong on the test (see discussion in 
Session 5). I coached Mulder as he applied this logical classification and linking process 
to the definitions he had prepared, modifying them in ways that made more sense to him 
or that were necessary to be accurate. I encouraged him to link this process with the use 
of the statistics they represented in the computation but he seemed bent on keeping the 
sections separate in his mind. 



288 

We briefly discussed his plans to complete his test preparation during the day, and 
he left to do his chores and go to work- 
in this session, I pushed Mulder to make logical decisions and connections and he 
quite vigorously resisted, using visual memory and pattern finding of generally 
extraneous details as benchmarks for decision making rather than exploring the logic of 
the material. 

The only thing Mulder had done differently to prepare for the multiple-choice 
section had been what we did in Session 3; he had not used the practice multiple-choice 
questions I had given him to do at home. He had written out defmitions for the symbols 
but his resistance to changing his approach or to doing more than memorize patterns and 
procedures remained entrenched. The resistance may even have grown and Mulder 
seemed to use considerable amounts of energy for this resistance. The more I reacted the 
more he resisted. I urgently needed to understand this resistance and help Mulder fmd a 
way to put his energy and intelligence into preparing for and taking his exams. 

I was getting frustrated! I experienced sessions with Mulder as enjoyable. Even 
when he was tired and grumpy, I found him quick and funny. I tried laughing at his 
outrageous strategies, appealing to his intelligence, scolding, and cajoling him but 
seemingly to no avail. I was acting out of my countertransferential role of mother of a 
rebellious teenaged boy. I tried but could not manage to get him engaged in anything 
other than sparring with me. He was certainly not interested in addressing his issues 
seriously while I was trying so hard to change him. 



289 

Supervision 

By the time I presented this case for supervision the results of Exam #3 were 
known. Mulder had earned a 76% but with the in-class open-book extra credit assignment 
he earned 5 of a possible 6 points his final grade for the Exam was 81%. He had 
significantly improved his symbol identification score but had lost 18 points on the 
multiple-choice section, 59% correct (on Exam #2 he had lost 14 points). In addition 
Mulder had lost 6 points on the computational section, most because he had failed to set 
his |Xd= 0, despite our discussion of this in Session 4 the morning of the exam. 

I presented Mulder to Dr. P. and realized that what I wanted to do was to talk with 

Mulder about how he stands in his own way and to propose to him that he had the choice, 

that he might be able to make the choice to stop doing it. Dr. P. suggested instead a 

strategy established by Alfred Adler (Mosak, 1995) and called paradoxical intention by 

Victor Frankl (1963) that might help Mulder make that choice. The theory suggests that, 

"The symptomatic patient unwittingly reinforces symptoms by fighting them... to halt this 

fight, the patient is instructed to intend and even increase that which he or she is fighting 

against" (Mosak, 1995, p. 74). Dr. P suggested that in the next session I have Mulder 

experiment with truly resisting on an exam. His directions were clear: 

Ask him how he resists; suggest that as an expert in resisting that he let me know 
what strategies he uses to do that, so that he paradoxically really exaggerates this 
thing; it's his life but as long as he is into resisting he might as well do it really 
well. (Dr. P., July 20, 2000) 

In presenting other cases for supervision (cf Brad and Autumn) I had revealed my 

tendency to tell rather than ask participants how they might make helpful changes using 

the insights we uncovered. Dr. P. gently but firmly helped me recognize how 

counterproductive that was. I needed to see how by my telling Mulder I was likely 



290 

exacerbating his resistance — he was not only resisting a conceptual understanding of the 
mathematics, he was resisting me as a "teacher" and perhaps even a "mother." 

Dr. P. also suggested that I have Mulder explore the implications of the metaphor 
that he was standing in his own way. "What's he doing and then what's the part of him 
that's standing in his way doing? What gesture? What sound? What stance? Is he 
tripping himself over? Is he holding himself back?" (Dr. P., July 20, 2000) 

Finally he suggested I compliment Mulder on his insight into how he gets in his 
own way, "Insight saves you a lot of trouble, not having to say, 'I don't know what's the 
problem here'" (Dr. P., July 20, 2000). I puzzled over this. What exactly did Dr. P. mean? 
In some ways I felt that Mulder might be expending too much energy struggling for 
insight (or was it an excuse?), so I had to concede achieving insight should certainly free 
that energy for actually doing the coursework. And the importance of congratulating him, 
mirroring his achievements in insight and cognition was becoming clearer to me. 

Session 5: Honing Resistance Strategies 
I had videotaped the class on the evening before Mulder's and my Session 5. 
Mulder sat beside and around the comer from me and was very interested in my research 
activities. He noticed that I was observing students and taking notes when no one else 
was writing. He "acted up" during the problem- working session in class to the extent 
that, at one point, I called him a "scallywag." He seemed pleased. 

At the start of tutoring I told Mulder I had found out more about Fox Mulder and 
he gave an appreciative laugh. I said that Fox Mulder seemed to me the kind of antihero 
who does things opposite to how others think they should be done. I asked Mulder 
whether that was the characteristic that appealed to him and he agreed that Fox Mulder 



291 

was like that, doing everything in a way that was "definitely indirect" but he denied that 
was the element he identified with. He insisted what appealed to him was Fox Mulder's 
constant effort in looking for the truth. It seemed to me that he was denying a reality that 
was largely unconscious (opposition) while consciously espousing a desire that was not 
yet a reality (fmding truth). 

I suggested Mulder's own search for "why I make mathematics so difficult for 
myself might entail exploring his underlying and seemingly growing resistance to 
mastering the conceptual part of the course (that Ann tested using multiple-choice 
questions) that seemed to be making it harder for him to succeed. Paradoxical intention 
theory suggests that getting Mulder to consciously and vigorously resist as he answered 
multiple-choice conceptual questions (i.e., enacting the very behavior he needed to stop) 
should result in his overcoming his resistance. 

I proposed to Mulder that he take a mini-test of conceptual multiple choice 
questions from the textbook's study guide'^' that I would give him, but that he should 
strongly resist while talking aloud about his resistance. He seemed intrigued but initially 
challenged my instruction to really resist with, "What, not do it? I can do that!" Mulder 
began working silently so I asked how he was doing. Rather defiantly he replied, "You 
tell me!" but in a few more minutes he said, "I don't like this one" and to my query, 
"Because I don't know if it's really, really easy or if I'll have to do some work to fmd it." 



292 

I began talking Mulder through the problem by tapping into his existing 
knowledge of the process of finding critical values using tables in the appendices in the 
back of the book. As we did this I probed a little, "Is that part of your resistance because 
you think about 'is that easy or is that hard?'" Mulder conceded, "Probably." 

He decided the answer to his original query (is it easy or will it require work?) 
was, "Too much work for me!" so we talked about how much work he was doing 
comparatively for the fmite mathematics class. When I asked if it was about the same as 
for the statistics class he demurred, "I don't know; next to 'bout none." I pursued this 
further since he had clearly been doing better in the fmite class from the begiraiing 

Mulder insisted it wasn't because he was doing more work or because it was 
coming more easily to him but rather that, "I just don't have to do conceptual questions!" 
I wondered, "Maybe then, [for] your resistance you say, 'This is conceptual. I don't have 
to do that.' Maybe if you could say, 'Ah this is not conceptual' Rename it: this is just 
mathematical. . ." Mulder blurted out "Pain in the butt!" I was prescribing and telling so 
his resistant reaction to me should not have surprised me yet I was startled and asked if 
he was calling me a pain in the butt but no, he insisted, it was "that section of the test." I 
responded, "Well you are doing a nice job of resisting, which is good!" I could not 
consciously acknowledge to myself that it was almost certainly me he was calling a "pain 
in the butt" and resisting. 

Mulder went on with his multiple-choice mini-test. He grumbled as he went, at 
one point exclaiming "Crap!" when he picked the wrong value for N. I guided him to 
interpret the table using the given a value of .01 and when he did it correctly I 



293 

commended him. He interpreted that as condescension on my part declaring indignantly, 
"I can readV I came back with, "You can also think!" 

I gave him the package of class problems with the test name whited out, intending 
him to use those after he was done with the multiple-choice, but he immediately grabbed 
them and began naming the test using the same type of pattern memory "logic" that he 
had exhibited in Session 4. As he expected, I scolded him reminding him of all the 
multiple-choice questions he was now avoiding and commenting again on how well he 
was resisting. 

At that point, I interpreted what I saw happening between us from my point of 

view, "I'm thinking this guy is so smart he could do so well and the mother in me comes 

out and it's like 'If I could only persuade him'" I spoke from my side of our power 

struggle. Mulder continued with the next question, appearing to ignore me. In a few 

minutes as he began to work on new questions, I commented. 

Every now and then you stand up against your own resistance by speaking [the 
resistance, but then you stop resisting and do the work]... First of all you say I 
don't want to do this, I don't want to come back here, she's a pain in the neck 
...but then [you see] what happens when you try them [the questions]. (Session 5) 

After this interchange I felt a distinct change in Mulder's demeanor. It felt like a 

turning point. He began to ask "why" questions. For example, "I have a question: Why do 

we find the total of the degrees of freedom?" Mulder became actively engaged in the 

discussion of my responses. When I admired his thinking he was spurred on, but when I, 

drew his attention to this apparent change and put it in terms of his forgetting about his 

resistance he became defensive and seemed to return to it. It would have been better if I 

had asked him "Does this feel different to you?" rather than telling him of the change I 

felt. By not doing so I was pulling him back into our power struggle. 



294 

I wondered out loud if his resisting was really letting the instructor win, a form of 
"I can't work this through... I'm wimping out here." Mulder ignored me and went on 
working, even suggesting that he do some from chapter 1 8 and triumphant when he got 
those correct. 

I went further with the Mulder-has-two-parts theory. I reminded him of his 

reaction to losing points only on the conceptual/symbol definition section in Exam #2. I 

suggested that he had seemed almost proud that he had proved the pomt that he couldn't 

do the multiple-choice conceptual part," but he disagreed, "Not proud of that just pleased 

I got all the calculations right . . .1 got a B!" He sounded indignant that I didn't seem to 

appreciate that fact and admire his competence. I wanted to give him a little more 

feedback. 

There's a part of you that is really listening and engaging in it and there's another 
part that's like "uugh" [pushing away with my hand]. It's almost like you've got 
this little battle going on [Mulder chuckles]. You don't think that's happening? 
(Session 5) 

Mulder continued with his mini-test as I looked at what he had done on Exam #3. 

I continued to alternate between exploration of Mulder's mathematics approaches and his 

resistance. I interrupted him to ask him about an error on Exam #3. He said that forgotten 

some details that Ann had told him so he made something up. I had also gone over this 

fact and the logic of it with him in Session 4 the morning before the exam, but he was at 

the time resisting my push to link symbol defmition to the computation. He clearly didn't 

recall that we had discussed it. Mulder had however successfully used his visual 

memorizing of in-class worksheets on parts of the exam He described the process, "That 

one I just sat there forever and ever until I came up with it. ...I kept thinking of the 

worksheet in my head until I came up with it." I wondered about what might have 



295 

happened if he had sat for ever thinking about the multiple choice questions. I suggested, 
"It's like a cooperative part that does this [the computational part] and a resistant part 
[that doesn't do the multiple choice]" 

Mulder continued working, talking out loud as he did liberally sprinkling his 
reactions with "crap" and "turd,' and at one point calling himself an idiot for not dividing 
accurately. He seemed intrigued with the concept of his two conflicting parts doing battle 
even though he said, "Let's not remind me!" when I noticed that he was doing lots of 
good thinking so "the cooperative side of Mulder is working." At one point when he 
called himself an idiot I was wondering out loud about his sense of his own competence, 
which he thought was fairly robust, and I hesitated thinking I was interrupting his work. 
But I was surprised when he said, "Keep going, I'm listening," so I did: 

JK: In the test when you're doing this part is there anything that your I-want- 

to-do-better; I-can-think-about-this; This-may-be-logical part of you can 
say to the resistant part that says "turd" or "I'm screwed!" or "I don't like 
this!" or "this is ridiculous!" 

Mulder: You quoting me still? You're going to write your dissertation and say this 
person said all this stuff about math. 

JK That could be part of it but the main thing is what you're saying to 

yourself (Session 5) 

Mulder sidestepped my question by drawing me into a difficulty he was having 

with x^ , but I felt that I was really being invited to witness his internal dialogue as he 

found his error and mastered the procedure and the concept — ^that was his answer to my 

question. Mulder was demonstrating his answer — that his smart, achieving self was in 

charge now. 

I became aware that in my counseling role, even my tutoring role, I should have 

taken a neutral stance with respect to Mulder's warring parts. I was truly moving closer to 

that but I was so used to takmg sides that I continued to betray my partiality for Mulder's 



296 



"good" side, risking as I did, propelling him towards his "bad" side and getting us back 
into our power struggle. 

After Mulder successfiilly negotiated more questions on x^, I moved in again as 
tutor. I wanted to help him make more visual and logical connections: 

JK: Have you seen a picture of a x^? [JK draws a graph] 

Mulder: So it's going to be positively skewed. . . 

She didn't show us this [resisting what he saw as additional material]. 

JK: [chuckles] That's true But it's not going to do her in if you get this wrong; 

it's going to do you in! 

Mulder: Yes. 

JK: Yes and you want to be done in to prove your pomt. 

Mulder: Yes that's my goal in this test: to fail miserably on the multiple-choice. 

JK: To prove your point. 

Mulder: So I have to take that fmal. 

JK: Ah, there you go! 

Mulder: I'm not going to take it if I don't have to. . . if I could do well on these next 
two tests I wouldn't have to take the [optional] fmal. 

JK: There you go! This is that logical sensible person. Don't speak to me! 

Speak to that resistant part that has to get a bad grade on the multiple- 
choice. (Session 5) 

Mulder continued, alternating between resistant grumbling and engaged 
cooperating. But the focus of his grumbling was changing. It was less about her [Ann] 
and having to do the conceptual work, and more about the cognitive demands of the 
conceptual work. He grumbled about a change in how to identify significance required in 
a new test, about a question we both agreed was badly worded in the study guide, and 
especially about the practice problems I had prepared for the exam. He tackled the rest of 
the multiple-choice questions in an engaged, positive way. And I stayed out of the battle. 

At the end of our session Mulder's fmite math instructor came in and Mulder told 
her in discussion "I have a resistant side of me." To her dismissive "Don't we all," he 
insisted, "I lost 18 points on the multiple-choice." This was the first time that Mulder had 
verbalized the theory of his two battling sides and owned it. 



297 

Final Discussion: Mulder and Mathematics Counseling 
Mulder completed the practice problems, spent time on his formula sheet, and 
even came to study group before the Exam #4 the next evening. And he earned a 91% 
overall and an 82.6% on the multiple choice section! He went on to earn an A on Exam 
#5, an A" on the MINITAB presentation, and a B in the class. He was satisfied with that 
and decided not to take the optional comprehensive fmal. He dropped in a couple of times 
at the Learning Center, once to have me check over his MINITAB presentation paper (I 
suggested corrections that he did not make because he did not have time.) and once to 
help him prepare for his finite math fmal. 

Session 5 was a pivotal session both for Mulder and for me. At the start of the 
session, I believed it was primarily his stubborn resistance against the assessment that 
prevented him from improving although I suspected that, in some ways, I was 
exacerbating that resistance by my mother-of-a-teenager countertransference. Though I 
planned to use Dr. P."s suggestion to try paradoxical intention, I found myself telling him 
what I saw of his resistance and suggesting how to fix it. Ah, that was it, I realized. When 
I tried to get him to do or know or believe something, he resisted me. His resistance to 
Ann and the conceptual part of the test was confounded with his resistance to me, and 
that in turn made me push harder. It was when Mulder and I assumed the same stance in 
looking at his approaches that the change in his self-awareness began. When I implied 
that he might be betraying or letting down his resistant self by the intelligent engagement, 
things didn't go well — I was taking sides. When I couched it in terms of two legitimate 
parts of himself that were engaged in battle, Mulder went with that and I was able to 
withdraw myself from the battle. I no longer had to battle Mulder's resistant side trying to 



298 

persuade him to capitulate and cooperate. He could fight the battle himself, using his 
intelligent engaged self, and I could let him go. It felt good (but a little scary) to pull 
myself out of the fight and let Mulder battle himself and fight his own demons. He did 
this successfiilly, and I congratulated him. 

Evaluations 

On the post-surveys, Mulder's learned helpless beliefs changed significantly 
towards mastery orientation (see Appendix M, Figure Ml and Appendix H, Table H3). 
He said his motivation for coming to mathematics counseling changed irom helping me 
with my research to getting help with his strategies because the help with the statistics 
was "great," but his metaphor had not changed much: the Truth (mathematics?) was still 
"out there." Mulder had found that despite a relatively underdeveloped, vulnerable 
mathematics self, he could do well in a mathematics course if he got out of his own way 
and tried to think strategically and conceptually. That very success could contribute to the 
development of that self There was still room for growth but now Mulder might draw on 
this experience and risk trying to understand rather than using illogical alien approaches 
or overconfidently avoid trying. 

I had learned to attend to Mulder's transference of past relationships with teachers 
into our relationship as he alternated among confessions of laziness, pronouncements of 
his potential, and theories about what mathematics was that precluded the parts he was 
struggling with. I learned to attend to my countertransference reactions: I was the 
frustrated, cajoling mother of a young man who seemed to be his own worst enemy, and 
he sparred with me and appropriately resisted my efforts to fix him. Mulder's own 
metaphor was the key to resolving the conflict. In supervision with Dr. P., I began to 



299 

understand better what was going on as we examined Mulder's metaphor and I shared my 
transference-countertransference insights. In our last counseling session Mulder resolved 
his central conflict when I withdrew from my countertransference stance. I truly used a 
relational counseling approach, the outcome for Mulder was good, and I learned how 
powerfiil and counterproductive countertransference reactions can be even when one 
theoretically knows about their reality. I learned that although it is difficult, examining 
my countertransference reactions and choosing consciously to do things differently is 
crucial. 

REFLECTING ON THE COUNSELING CASES 

Crossing previously drawn lines — that seemed to be a common thread through the 
course of mathematics counseling with Karen, Jamie, and Mulder. Indeed a relational 
approach required it. I crossed lines and so did they, and we crossed lines together. 
Although mathematics was our primary activity, my persistent curiosity into how they 
did well and why they struggled led us to new ground. Of the three Mulder was the most 
willing on the former and the most resistant on the latter. Karen put strict limits on her 
responses to what she perceived to be non-mathematical discussions but she crossed her 
own previously drawn lines in mathematics effort. Jamie was willing to cross lines with 
me after I crossed lines to draw her into counseling in the first place. But we stayed 
within boundaries acceptable in the Learning Assistance Center context. 

Each of these students made progress academically. Each achieved a grade as 
high as or higher than they had hoped (see Appendix H, Table HI .). Mulder and Karen 
who earned D" s on the first exam went on to earn Bs in the class. Both did so despite 
significant deficits in their mathematics preparation (see Tables 6.1 and 6.2). Jamie foimd 



300 



Table 6.1 

Focus Participants ' Levels of Understanding of the Variable on the Algebra Test 
(Sokolowski, 1997; Brown et al., 1985, p. 17; see Appendbc C) 



Participant 


Number 

Correct (of 

53) 


Level 1 


Level 2 


Level 3 


Level 4 


Level 5 


Level of 
Understanding 


Jamie 


42 




6/6 


5/7 


8/8 


6/9 


1/3 


Level 4 


Karen 


30 




6/6 


5/7 


3/8 


2/9 


1/3 


Level 2 


Mulder 


25 




6/6 


7/7 


3/8 


3/9 


0/3 


Level 2 



that, contrary to her belief, she was adequately prepared mathematically (see Tables 6.1 
and 6.2). She and Karen were repeating the class and they saw and did things differently 
and did well this time. 

And each of these students gained new insights into themselves as mathematics 
learners. Jamie realized that her difficulties with mathematics were not to do with her 
ability but rather with relational issues; Karen found that she could achieve well in 
mathematics despite her considerable arithmetical and algebraic deficits and lingering 
doubts; and Mulder overcame his resistance to aspects of mathematics he found difficult 
because of his auditory processing difficulties and in defense of his vulnerable 
mathematics self and found that he could do well. 

I crossed lines and found a new way of looking at myself and them and us that 
gave me new power to reflect, monitor, and change my approach and steer the 
counseling. At the same time, this new way of looking gave me new ways of listenmg, 
observing, and responding to them so that they could and did choose their way and 
modify mine. The counselor-student dyad indeed was the key to the changes we all made. 



301 



Table 6.2 

Focus Participants ' Understanding of Arithmetic on the Arithmetic for Statistics assessment 
(Appendix C and chapter 8 discussion) 



Participant 


Class 
work 


Small 
(<1000) 


Large 
Integer 


Fractional 
number 


Place 

Value/ 


Operation 

Sense 


Open Ended 
Arithmetical 


Statistical 
Sense 






Integer 


Number 


Sense 


Decimal 




thinking/ 






Number 


sense 




Sense 




problem- 








sense 










solving 




Jamie 


adequate 


100% 


43% 


67% 


|a|<5: 


80% 


95% 


88% 






adequate 


inadequate 


-adequate 


97% 

adequate 

|a|>5: 

100% 

adequate 


adequate 


adequate 


adequate 


Karen 


marginal 


45% of 


17% of 


33% of 


|a|<5: 


20% of 


35% of 


45% of 






total; 100% 


total; 30% of 


total; 


65% 


total: 33% 


total; 42% 


total; 






of attempts 
inadequate 




attempts 
inadequate 


75% of 

attempts 
adequate? 


-adequate 

|a|>5: 40% 
inadequate 


of attempts 
inadequate 


Ofattempts 
inadequate 


62.5% of 

attempts 
-adequate? 


Mulder 


adequate 


100% 


56% 


66% 


|a|<5: 90% 


100% 


76% 


56% 






adequate 


marginal 


-adequate 


adequate 

|a|>5: 69% 
adequate 


adequate 


adequate 


marginal 



good X>85% 

adequate: 70% < X < 85% 

-adequate: 60% <X < 69% 

marginal : 50% <X < 59% 

inadequate: X < 50% 

inadequate? X > 50% but < 50% attempted 



adequate?: X > 70%. ofattempts 

adequate?: 60% <X < 69% of attempts 



In the next chapter I briefly profile the remaining nine students in the class and 
discuss the developing theory that emerges from this pilot study. In particular I propose 
criteria for a new way of categorizing students as mathematics learners that surfaced from 



302 



analysis of participant profiles. I then analyze and present what I see as the essentials of 
the brief relational mathematics counseling approach that emerged. 



303 



' Because this is an even numbered chiapter 1 use "she," "her," and "hers" as the third person singular 
generic pronouns. 

" This pattern continued except for two occasions later in the course before exams when she did allow 
herself to be drawn in to some of the study group's discussion. 

■" I saw the impact of this limited understanding in the first exam when she used the deviation of only one 
score from the mean instead of the required deviations of all values of the variable (the scores), to find the 
standard deviation of all the scores fi-om the mean. 

" Performance vs. learning achievement motivation questions were numbers 4, 7, 9, and 10 of Part I of the 
Beliefs Swvey (see Appendix C) 

■" For example, she mistakenly thought o (sigma, the standard deviation of the population.) represented the 
mean of a population. 

" For example, \x = population mean, and a = population standard deviation, both constant identifiers for a 
particular population distribution. 

™ For ex2mp\e, X sample mean, and 5 sample standard deviation, each constant identifiers for a particular 
sample. 

"" She had to find the median, P50 , of a set (distribution) of scores. 

" She agreed with prompting that her answer didn't make sense but when I instructed her in the use of her 
calculator to get the correct answer she remained baffled, "I don't really know why though. I just plugged 
in what you told me." In this session there was no time to teach Karen the concept of percent. 

" Despite her emotional state in Session I, Karen had managed to successfijlly correct, learn, and retain 
these symbol designations during the session. 

'" For example, for the real limits of the weight of a slice of cheese of 0.35 grams, Karen had answered 

d. may be anywhere in the range of 0.34 -0.36, instead of the correct 

b. may be anywhere in the range of 0.345 - 0.355. 
Karen's answer to this and similar questions showed an understanding of the concept being tested but 
anything beyond the first place (tenths) of decimals confijsed her. Karen was able to give 8.5 and 9.5 as the 
real limits for 9 and other whole numbers but not for 0.9 or 2.9. She misnamed decimal places, calling 
hundredths tenths and vice versa. She did not seem to have a firm sense of the relative size represented by 
the places nor the places' relationships with each other. 

"' In reality, when Karen took the Arithmetic for Statistics assessment after the course ended, she showed 
that she c/zii understand relative sizes represented by the places, on the number line graphing questions 
though not by using numerals alone. If Karen had made that extra appointment, I would have given her the 
Arithmetic for Statistics assessment then and coached the exploration beginning with her number line 
understEinding. 

™' We each copied the data fi-om the class question onto our own sheets of graph paper. Next we copied the 
formula as I read it out loud and constructed columns beside the X and the Y columns with column 
headings corresponding to pertinent elements of the formula for /% namely X", Y", and XY. 

'"" I suggested we use the questions for the assigned problem in the text (Pagano, 1998, chapter 6, Problem 
14, p. 124) and the data from the in-class question because the questions were more delving and the data 
were less complex. 



304 



'" Karen had not computed the standard error of estimate of Y accounted for by X correctly because she 
had not constructed the squared deviations (Y -Y'f column. She had the (Y -Y') but not the (Y -Y')^ 
column on her formula sheet, so she used the (Y -Y'). This should have summed to zero prompting her to 
check her formula (which correctly included the squaring) and create the (Y -Y")^ column. Instead of 
addingKaren tried multiplying the (Y-Y')s to get 0.00015 Iwhich she then wrote as 1.51 tomakeitmore 
reasonable. She knew this was not correct, however, as she wrote on her exam "still can't figure out where 
I screwed up." 

'^ I had checked with Ann about student use of the blank flow chart and she was agreeable. 

""' Karen knew from class that being given 5" s or the SSs (sums of squared deviations of scores from the 
mean) would indicate an independent samples t test in contrast with two sets of data that would indicate the 
correlated groups t test. / knew, however, that in the real world of data gathering and analysis, students 
could be given two sets of scores for eitlier situation, independent or correlated, and coi/W calculate SSs and 
r s from that data for either. 

™" Karen made an error in one inference test that did not result in her losing points. On the normal deviate 
z test, Karen had compared the magnitude of the p value she obtained (0.0013) with the z score she had 
obtained (-3.01) instead of with the critical alpha level of 0.05. She came to the correct conclusion though 
so Arm did not deduct any points. 

'"" Still later, a semester after the end of summer PSYC/STAT 104, however, Jamie revised her stated 
motivation to helping me: "...I'm much more of helper, which is why I think I signed up to do this with 
Jillian, cause 1 saw it as helping her with her project. If it had been just for my benefit I don't really know if 
I would have approached her or not." 

"" Jamie was, in fact, repeating this course, but I did not find that out until after the course was over. With 
appropriate permission, I obtained the printouts, without names, of the grades of all students of the 
PSYC/STAT 104 for the 5 years before the summer of 2000. It included the data from this class with some 
that suggested that Jamie was repeating the class. When I sent a post-study e-mail survey to check that and 
other data 1 was unsure of, Jamie replied to my assumption that she was not repeating the course, that in 
fact she was repeating it because she got a D* the first time and that was not adequate for her psychology 
major. 

^ Jamie's "?" indicated her own uncertainty about her exact grade. 

'"' She did not tell me of her D* in her first attempt at PSYC/STAT 104 at State University. 

'™" See Chapter 5 for a discussion of Ann's policy regarding students' tests. 

'""" Here Jamie indicates her belief that her error lay in using "the right one [equation] for a different one" 
perhaps thinking of the different formula for a, the standard deviation for a population for which the 
denominator is N rather than the n - 1 for the s, the standard deviation or a sample. In feet, the class had 
learned no formula for which the denominator is EX - Ithe one she ad initially used. 

"^ 3.7 on a 1 through 5 scale — close to the highest in the class (see Figure L4, Appendix L) 

'°™ Sokolowski's three college student subjects who achieved a level 4 of the algebraic variable, had each 
succeeded in at least one college level mathematics course, was at the time of her study an A/B 
mathematics student, and succeeded in combinatoric-/probability-/statistics-related mathematics (p. 70, 98). 

'°™" Since a score of 1 represents zero anxiety, the drop of 0.5 in Jamie's Mathematics Testing Anxiety 
form 4. 1 represents a 0.5/3. 1 that is a 1 7% decrease. 



305 



'°"™ This participant chose this pseudonym for himself when the question arose during his mathematics 
counseling Session 5 on July 25, 2000. Fox Mulder was also his metaphor for how he approached 
mathematics. 

""" Over the course he talked during the lecture portions of the class in every class but one, averaging three 
interactions — answers or corrections — per class. This placed him as the third most involved in these lecture 
discussions, after Robin and Lee (see chapter 5, Table 5.2). 

"" A science fiction television series featuring FBI paranormal detective Fox Mulder (and his partner 
Scully) in search of the aliens who he believed had abducted his sister. 

"""^ I was able to check on this more formally when I gave Mulder my Arithmetic for Statistics assessment 
(Appendix C) as a posttest on July 31, 2000. He asked if he could fill it in later and eventually sent it to me 
in March 2001 . See Table 6.2 for Mulder's results, all of which were adequate except for his statistical 
number sense and large integer number sense which were marginal. These last areas (tested on this 
assessment) were not tapped during the course. 

'°°"' His approach lacked a number of the identifying features of learning style II; he did not seem to grasp 
the gestalt of a situation or use an inductive (rather than deductive) reasoning approach nor did he have 
difficulty with details and step by step procedures. In these areas he seemed more analytically procedural 
(like Davidson's mathematics learning style I) though he used visual memorization rather than verbal 
tactics. On the other hand his finding solutions without being able to satisfactorily explain how and his 
sense of appropriate sized solutions supported a learning style II conjecture. It was also not clear to what 
extent he had adapted his approach to handle mathematical tasks that seemed beyond him. 

■"^ Suinn (1972) found on his 98 item Mathematics Anxiety Rating Scale (MARS) from which all the 
testing and number anxiety items of my Mathematics Feelings survey are drawn, that mean scores were as 
low as 1 .47 for physical sciences majors (sd = 0.4), and 1 .7 for social sciences students (sd = 0.6) which 
would seem to imply that Mulder's 3.1 shows high anxiety (more than 2 standard deviations above the 
mean). But because Suinn's scale was found to confound testing and number anxiety factors (see Rounds 
and Hendel, 1980) and students' number anxiety scores were on average 0.75 points lower than their 
testing anxiety scores when separated here in the Mathematics Feelings survey, 1 would suggest a higher 
average for testing anxiety and a lower average for number anxiety than Suinn's should be considered 
moderate on my Feelings survey. Given this consideration Mulder's 3.1 testing anxiety score could still be 
considered well above moderate even for a social sciences student. 

'""""The fact that he was 75% satisfied with his mathematics achievement (Item 6, responses marked 1) and 
75% confident about his mathematics future (Item 3, responses marked I), and 75% positive about the 
course he was taking now (Item 4, responses marked 1 ) indicated a positivity that did not seem justified by 
his history or his performance on Exam #1 (see Appendix M, Figure M2 and Table M3). 

'°°" That is, to have him discover that developing a conceptual understanding of the procedures he had 
mastered by Exam #2 would help ensure continued success in the computational part and mastery of the 
conceptual part of the test. 

""^ He pointed to question 3. 

3. The 20 subjects constitute a 

a. population 

b. sample 

c. parameter 

d. variable 

Mulder asked me who the 20 subjects were; what was that about? 1 found that he had not realized that the 
first 5 questions were referring to an experiment described and bolded at the top of the page. He got three 
of these five questions wrong (and lost 6 points). 



306 



'"""' Ann mislaid the computational section of Mulder's first exam so we were unable to analyze his errors 
on that section. 

'°°™" Mulder wanted to "memorize rules for doing a SIMPLEX problem." He felt he had the material under 
his control and was really just checking that he had it correct. He knew the material procedurally but was 
not able to explain to me nor did he want to know why he had to do what he was doing. He was using the 
SIMPLEX method to maximize profit given a system of linear constraints (Rolf, 1998, chapter 4). He knew 
that an equation had to be changed to two inequalities, in particular, inequalities in which the variable sum 
was less than the constant, before slack variables could be added, but he did not know why, probably 
because he did not understand the meaning and use of the slack variables. He was able to perform the 
necessary procedures. 

'°°™' For example, Mulder thought 5 stood for "sample" (rather than sample standard deviation). 

"^ For example, we worked out what D meant knowing already that X was the mean of scores for a 
sample. 1 pointed out the links: the bar conveys the idea of mean and the D represents the list of data being 
analyzed (in this case the differences between pairs of before and after scores). 

^' The chapters covered were: Chapter 15: Introduction to Analysis of Variance, chapter 16.' Multiple 
Comparisons and chapter 1 8: Chi-Square and Other Nonparametric Tests in Understanding Statistics in the 
Behavioral Sciences {Pagano, 1998). 



307 

CHAPTER VII 

DEVELOPING THEORY: STUDENT CATEGORIES AND 
WAYS OF COUNSELING 

In this study I gathered mathematics cognitive and affective data from 12 of the 
students of PSYC/STAT 104 and I counseled ten of them using cognitive constnictivist 
tutoring and relational and cognitive counseling approaches described in chapters 2 and 
3. The results of the study are described in chapters 5 and 6. As I analyzed these results I 
noticed a number of interesting interlocking patterns that I discuss in this chapter. I will 
demonstrate how this analysis supports a categorization scheme of mathematics learners 
that emerged from this research. I will then present my analysis of this brief relational 
counseling approach as I found it relates to students thus categorized. 

When I analyzed the three in-depth cases and the briefer profiles of the other nine 
students in the class I found that categories of mathematics self development emerged 
from interactions between two dimensions — mathematics preparation and relational 
experience. These interacting factors produced relatively well-defined categories that can 
be compared and contrasted with Tobias's tiers described at the end of chapter 4. These 
categories although similar to Tobias' tiers are distinct in important ways. 

This result was of particular interest because I also found that different relational 
mathematics counseling approaches and the relative balance among its components 
(degree of cognitive constructivism, amount and kind of mathematics tutoring, amount of 
course management counseling, and cognitive and relational counseling) were differently 
applicable to specific categories of student. 



308 



Mathematics Preparation 

Students in the class fell into three broad categories according to the adequacy of 

their mathematical preparation for the class: well prepared, adequately prepared, and 

underprepared (see Table 7.1). 

Table 7.1 

Criteria for Determining Level of Mathematical Preparedness of PSYC/STA T 104 
Participants 



Well Prepared 



Adequately 
Prepared 



Underprepared 



Course grades: 








Exam #1 


B+ through A 


D through A 


F through C 


Final Course Grade 


A' (B^) through A 


B" through A~ 


AF through B 


Algebra: 








Algebra Test 


Level 4 or 5 


Level 4 


Level 1 or 2 (or 3?) 


and class, exam and 








counseling session 








work 








Arithmetic: 








Arithmetic for 


Good (>85%)in all 8 


Adequate or above 


Ranges from 


Statistics 


categories 


(>70%) in all but 


adequate or above 


Assessment and 




one or two number 


(>70%) on at most 6 


class, exam. 




or operation sense 


categories to 


counseling session 




categories; variable 


inadequate (<50%) 


work 




in other sections 


or marginal (50% < 
X < 59%) on three 
or more categories 



Evidence for how participants placed in these categories was gathered throughout 
the course. Not all participants took the Algebra Test (Robin, Brad, and Kelly did not.) or 
i\\Q Arithmetic for Statistics Assessment (Robin, Mitch, Brad, and Kelly did not.) but in 
these cases there was sufficient evidence from their exams and work in class and 
counseling by the end of the course to place them with reasonable confidence. The three 
criteria that served best to categorize students in this sample were (a) understanding of 
the algebraic variable (measured on the Algebra Test, see Appendix C), (b) understanding 



309 

of and facility with arithmetic (measured on the Arithmetic for Statistics Assessment, see 
Appendix C), and (c) performance on the first exam of PSYC/STAT 104. When I 
considered students' high school and college course-taking and grades as an additional 
criterion for this sample, there was not enough consistency for this to be usefiil (although 
with larger groups of students this might be found to be a factor). The students who 
were well prepared rwAhemdiiic&Wy had a high level understanding of the algebraic 
variable (see also Appendix H, Table HI), were arithmetically confident and competent, 
had always done well in mathematics, and did well on the first exam in the course (see 
Table 7. 1). Those who were adequately prepared had a high-enough level understanding 
of the algebraic variable. However, while their arithmetic was generally sound they had 
some deficit areas, they had each had variable success in previous mathematics courses, 
and their performance on Exam # 1 ranged widely from D~ through A. Those who were 
tinderprepared had a low level understanding of the algebraic variable, deficits in 
arithmetic that ranged fi"om significant to mild, and they did poorly on the first exam of 
PSYC/STAT 104. 

It is possible that using this approach to classifying students with larger groups 
might result in the imderprepared group' s being split into more categories. With this 
small group, placing the student/s who were weak in both arithmetic and algebra in the 
same category with student/s who were weak in algebra but sound in arithmetic makes 
sense given other identifying criteria. Further evidence may suggest otherwise. 
Mathematics Self: Mathematics Preparation and Self-Esteem 

After I sorted students according to their mathematics preparation (see Table 7. 1), 
further analysis revealed that students' level of mathematics self-esteem roughly matched 



310 



the preparation categories and that these taken together gave a measure of students' 
mathematics self development. As noted in chapter 2, self psychologist, Kohut (1977) 
proposes that healthy self development leads to internalized values and ideals that 
provide structure and boundaries as the person's own competence develops. When this 
process proceeds appropriately the internal self-structure is consolidated and it provides 
what Kohut calls "a storehouse of self confidence and basic self-esteem that sustains a 
person throughout life" (p. 188, footnote 8). From this study I found that it was a 
student's mathematics competence (preparation) taken with his level of his self-esteem 
that indicated that self development level: Category I (sound), II (undermined), or IE 
(underdeveloped) (see Table 7.2). I found that his level oi confidence . realistic, under, or 
overconfidence, was an initial cause of confusion in assessing a student's category of 
Table 7.2 
Emerging Categories of Mathematics Self Development 





Mathematics Preparation 


Level of self-esteem 


Well Prepared 


Adequately 
Prepared 


Underprepared 


Sound self-esteem 


Category I students 

with sound 
mathematics selves 






Compromised self- 
esteem 




Category II students 

with undermined 

mathematics selves 




Low self-esteem 






Category HI 

students with 

underdeveloped 

mathematics selves 



mathematics self (see Ja/w/e, Karen, and Mulder in chapter 6). I found that a student's 
level of self-esteem, however, was directly related to his mathematics preparation 



311 

(competence) level. The levels of self-esteem I found in students in this sample were: 
sound, compromised, or low (see the student profiles below for discussion of how I 
discerned these levels). The shaded cells in Table 7.2 indicate that I found, as I expected, 
no student whose level of self-esteem was not directly related to his level of mathematics 
preparation. 

Mathematics Self Category and Relational Malleability 
I found that students in the second and third categories of mathematics self could be 
further sorted according to the extent of malleability (willingness to change beliefs and 
behaviors) versus inflexibility (resistance to changing beliefs and behaviors) in their 
mathematics relational patterns. This malleability versus inflexibility seemed to stem 
from personal characteristics interacting with past mathematics experiences in the current 
course environment. Students in Category II fell into these two subcategories according 
to how they had handled their compromised self-esteem: they had developed mathematics 
relational patterns that were either malleable or inflexible for the brief semester 
timefi"ame. Students in Category 11 of mathematics self similarly fell into these two 
subcategories according to how they handled their low self-esteem (see Table 7.3 for 
criteria I used to gauge malleability). This classification became important from early in 
the course because a student's willingness to engage in the struggle early in the course 
and to change if he' was persuaded that he needed to was, not surprisingly, a pivotal 
factor in his success. This was especially important for underprepared students with low 
self-esteem (i.e., students with an underdeveloped mathematics self). 

Interestingly, students I found to be inflexible seemed to fit Tobias' categorization 
of students as "utilitarian" (see chapter 4) and Mercedes McGowen's categorization of 



312 



Table 7.3 



Criteria for Determining Malleability of PSYC/STA T 104 Participants 





Malleable Relational Patterns 


Inflexible/unstable Relational 
Patterns 


Achievement motivation" 


Stated learning achievement 


Stated performance achievement 




motivation /learning motivation 


motivation 




when he believes he is capable 




Learned Helpless versus 


Mastery orientated in beUefs 


Learned helpless in beliefs and 


Mastery Oriented 


and/or behaviors 


behaviors 


beliefs''/behaviors 






Procedural versus Conceptual 


Conceptual mathematics beliefs 


Procedural mathematics beliefs 


Mathematics 


and/or behaviors and/or change 


and behaviors 


beliefs%ehaviors 


towards conceptual 




Problem-solving/trouble 


Engagement in problem-solving 


Avoidance of problem-solving 


shooting beliefs/behaviors 


practices/behaviors 


practices/behaviors 


Changes over course 






1 . Behaviors 


Willingness to change/ 


Resistance to change over 




resistance to change that 


course 




changes to willingness to 






change during course 




2. Beliefs: 






a) Mathematics' 


a) Substantial change in 


a)Limited positive changes in 




behaviors/beliefs over 


beliefs/behaviors over course 




course — some positive 






(especially focal beliefs or 






emotions), some negative 




b) Fixed trait beliefs 


b) little need for change or 


b) limited change 


about personality. 


positive change for this 




limitations, and/or 


situation at least 




mathematics 






potentiaf 






3. Attachment Patterns'* 


Secure; avoidant to secure; 


Remains detached or 




dependent to secure 


dependent or ranging between 
both 



Note: " Achievement motivation beliefs were gauged initially by averaging 1 through 5 responses on a 

subscale of the Beliefs survey: Part 1, Questions 4. 7. 9, and 10 (see Appendix C). Achievement motivation 

behaviors and fiuther explanation of beliefs were gauged through observation and conversation in 

counseling. 

'' These beliefs were gauged initially througli responses on the Beliefs subscale; behaviors and fiirther 

explanation of beliefs were gauged through observation and conversation in counseling. 

" Changes were gauged through posttesting oi Beliefs and Feelings surveys, by conversatioa and by 

observation of responses (verbal and behavioral) to counseling interventions and in course achievement. 

■^ Changes in attachment patterns gauged in counseUng through transference/countertransference etc. 



313 

students as "rigid" (personal communication, April 1 1, 2000). However, in addition, 
some malleable Category III students (e.g., Karen and Mulder) initially presented as 
inflexible (utilitarian/rigid) but their apparent inflexibility turned out to be defensive in a 
way that was adjustable with appropriate relational counseling. 

I am very aware that my own personality and behaviors might have been a factor 
in the extent to which a participant exhibited malleable or inflexible behaviors in the 
counseling situation. This may not be entirely intrinsic to him. My particular challenge as 
a counselor may be to develop ways of helping students I perceive to be relatively 
inflexible to bend. Effective ways to achieve that are certainly not by advising, lecturing, 
or scolding. A relational understanding led me to see that inflexibility may be at least in 
part in defense of compromised or low self-esteem. It was in students' responses to the 
surveys and their interactions in the classroom, with the course material, and with me that 
a malleable or inflexible profile emerged. Whether and how students changed over the 
course also helped confirm such a malleable or inflexible profile (see Table 7.3 for details 
of criteria and individual characterictics and Table H3 in Appendix H for student 
changes). No one met all the criteria identified for a profile but each participant had a 
predominance of characteristics of one type with relatively fewer of the other. 

I will now present brief profiles of the students in the study showing how they led 

me to develop the categories and sub-categories I have identified. 

Category I Students with a Sound Mathematics Self: Mathematically 
Well-Prepared with Sound Mathematics Self-Esteem 

In this class there were only two students who fit the Category I mathematics self 

profile and both had malleable relational patterns. It is possible that Category I students 

could exhibit inflexible relational patterns but I believe that to be unlikely unless such 



314 

students are faced with mathematics challenges well beyond what they are prepared for. 

Then the inflexible/malleable distinction might surface. 

Sound Mathematics Self Students with Malleable Mathematics 
Relationship Patterns 

Two students in the class (Catherine and Robin) each had a constellation of 
characteristics that identified them as mathematically well-prepared students with sound 
self-esteem: Each earned a good grade on Exam #1, had experienced prior steady success 
in mathematics and had no crucial knowledge base gaps in arithmetic or algebra, had 
more conceptual than procedural beliefs, showed mastery orientation to mathematics 
learning, and had low to moderate anxiety. They had learning (rather than performance) 
motivation for taking the course, and each was realistically confident and exerted a 
realistic amount of effort towards mastery in the current class. In other words, each had a 
well-developed mathematics self, no toxic internalized mathematics presences, and 
current patterns of mathematics relationship that were flexible and constructive. 

Although Lee had the highest conceptual beliefs score in the class on the pre- 
beliefs survey and had recently succeeded in a finite mathematics course in college, her 
high anxiety scores, low confidence about her mathematics (related to struggle and 
variable success in prior mathematics courses), relatively low Exam #1 score, and 
underdeveloped arithmetic operation sense ruled her out of this group. Autumn's 
performance motivation, procedural beliefs, learned helpless orientation, and history of 
uneven mathematics course performance also ruled her out of this group despite her high 
expectations and good grade on Exam #1 . Robin signed up for mathematics counseling to 
help me with my research, but Catherine declined the offer. 



315 

Catherine. Catherine," a non-traditional biology major who had just completed 
Calculus I with an A, was confident but quiet in class. Her high conceptual belief score, 
low anxiety, and high course outcome expectations seemed congruent with her presence 
in class and she was not considered to be at-risk in a statistics class. Initially, the 
instructor and I both thought she would do well without help. I wondered about how she 
would handle her own expressed need for conceptual understanding of the mathematical 
procedures {Beliefs survey) because these links were not generally made in class, but she 
did enough work on her own (5 hours per week) to make the conceptual links to the 
mathematical procedures that she needed in order to master the material. She did not ask 
for nor seem to need mathematics counseling. 

Robin. Robin's mathematics successes were much more distant in time, and she 
obviously struggled in class in both lecture discussions and during problem-working 
sessions. She was not initially recognizable as a Category I student. Her membership in 
one of the groups traditionally at risk for Brookwood (older, female, nursing'" students) 
and her classroom presence initially raised questions for both the instructor and me about 
her prognosis in the class. I had more early information about Robin than the instructor 
because of my pre-course surveys, but it was in such contrast to how she presented 
herself in class that I questioned its reasonableness. She seemed to need mathematics 
counseling so I was not surprised that she chose to participate — I thought she would need 
considerable emotional and cognitive help. It was in the counseling setting that I 
observed Robin's competence and confidence. I discovered then that she volunteered for 
the study to help with my research, and not because she believed she needed help. 



316 

Robin revealed her positive mathematical self-esteem and history through her 
metaphor for herself — Belle (from Walt Disney's animated movie Beauty and the Beast), 
an intellectually curious and competent feminine woman. She explained this in terms of 
her family's identifying her with a mathematical grandmother. Her success in school 
mathematics was tempered by her parochial school teachers' censure when she knew an 
answer but could not explain how she arrived at it. Robin seemed to be a global learner'^ 
with some auditory processing difficulties. She had not taken a mathematics course for 25 
years. 

My counseling support consisted of helping Robin become conscious of her 
positive mathematics self-concept by interpreting her Belle metaphor, inviting her to tell 
her story, and affirming her achievements and her current approach to mathematics 
course material. I chose parallel conceptual-to-procedural link tutoring to help her feel 
more grounded in her competent, conceptually oriented mathematics self Given how 
long it was since she had taken a mathematics course, Robin expended a realistic 10 
hours per week on homework. She struggled successftilly to compensate for her learning 
style challenges and make the necessary conceptual-procedural connections. Although 
the instructor's perception of her as a struggling nursing student never changed, Robin's 
confidence improved, she mastered the material to her satisfaction, and she earned an A~. 
Even without the mathematics counseling, it is probable that Robin would have done well 
but her mathematics base for further mathematical study became much more secure 
because of the affirmation of her good mathematical abilities by a mathematics expert. 



317 

Category II Students with Undermined Mathematics Selves: Adequately 
Mathematically Prepared with Compromised Mathematics Self-Esteem 

I found that more than one third of the students were adequately prepared to 

succeed in PSYC/STAT 104 but because of the interactions of past experiences with 

personal characteristics, they had developed relational patterns that could compromise 

their mathematics success in this class. They had relatively sound mathematics selves that 

had been undermined. Within this group there seemed to be two subtypes that I 

characterized as: a) students with malleable mathematics relational patterns and b) 

students with inflexible mathematics relational patterns (see Table 7.3). Students from 

these subtypes seem to have reacted differently to similar assaults on their developing 

mathematics selves. 

Undermined Mathematics Self Students with Malleable Mathematics 
Relationship Patterns 

The students who fell into this group were Lee, Pierre, and Jamie. They saw 

themselves as successful students in all but mathematics (and perhaps the sciences). 

They either underestimated or were ambivalent about their mathematical ability because 

of mixed mathematics success in the past. This caused moderate to severe affective 

problems in the Ann's mathematics class, particularly anxiety (for the women) and an 

expectation that they might do worse than they hoped. Generally, they had sound 

algebraic and arithmetical conceptual understanding but each had important gaps. Their 

beliefs about mathematics ranged from slightly more procedural than conceptual to 

conceptual, and they responded with positive mastery orientation to the challenge of 

developing a conceptual understanding once they believed they could. In other words. 



318 

each had an underlying sound-enough but undermined mathematics self from which he 
felt separated. 

These 3 students signed up for mathematics counseling with an initial motivation 
of getting help to negotiate the course. Although it became clear that each had a good- 
enough mathematical knowledge base"^ to succeed in this course (despite variable Exam 
#1 results, see chapter 5, Table 5.1), each had secondary problems that could have 
jeopardized this success. The women had developed anxiety problems expressed in their 
Feelings survey responses that were confirmed by observation and in discussion; the man 
had developed over-inclusive study practices that were counterproductive. Whatever the 
complexity, these students were willing to change their course approach in order to 
understand the concepts and achieve good grades. 

Lee. From the beginning, Lee was the most mathematically insightful of the 
participants. She was interested in how different elements of statistical analysis related to 
each other (see chapter 5, discussion of Study Group 1, pp. 175-176). She was the second 
most verbally responsive student in the class, with an average of 3.36 questions or 
answers per lecture discussion. Most of her questions were about exam strategy and 
concepts. Lee initially had the most conceptual beliefs in the class and was significantly 
more mastery oriented than learned helpless (on the Beliefs survey), but all three of her 
anxiety scores were high; her testing anxiety and number anxiety were each the second 
highest in the class and her abstraction anxiety the third highest on the Feelings survey. 
She signed up for mathematics counseling the day before the first exam because of 
anxiety, but we could not meet until after the exam. She had blossomed in a mathematics 
environment where she was required to think and explore deeply. She was driven to the 



319 

point of anxiety in classes where the conceptual connections to the procedures were not 
explored and where she felt that only mathematics procedures were being taught. 
However, she was convinced that, because the mathematics was not immediately clear to 
her and she had to work hard to understand, she was not good at mathematics. 

Lee found the PSYC/STAT class difficult because of a lack of in-class guidance 
linking concepts to procedures. She did well with the instructor's problem- working 
approach because it forced her to explore and master the procedures herself It seemed 
that she did not feel secure in her relatively sound mathematics self because of variable 
past success in her past and her self-comparisons with peers who "just got it" without 
having to work hard at it as she did. Lee's strong performance on the Algebra Test and 
Arithmetic for Statistics assessment helped allay her concerns somewhat (see Appendix 
H, Table HI), but she performed poorly (< 50%) on the operation sense section of the 
Arithmetic assessment. This significant gap seemed to affect her mathematics self and 
probably contributed to her anxiety. 

In our sessions I focused on affirming Lee's conceptual problem-solving 
orientation and providing a secure base for her to explore the concepts and the 
cormections that she did not experience in class. Lee relied on these sessions perhaps too 
much. She reported at the end of the course that she did only about 20 minutes homework 
a week. That was likely a factor in the high testing anxiety that increasing over the 
course. 

Another issue in Lee's anxiety may have been linked to the fact that she valued 
the conceptual understanding of the mathematics but may have undervalued the 
importance of thoroughly mastering the procedures. Her grades fluctuated, apparently 



320 

linked to whether she and I practiced the mathematical procedures or not, but she finished 
the course with an A after taking the optional comprehensive final to replace a lower test 
grade. 

Mathematics counseling was beneficial for Lee. It provided a secure conceptual 
base so she could repair her undermined attachments to mathematics and supported her in 
making the conceptual links to the procedures. In mathematics counseling I should have 
given more attention to providing bridges of understanding between her and her 
instructor (given their different priorities). I did continue to affirm her sound ability, 
learning motivation, and mastery orientation to achievement tasks, and Lee became more 
mathematically self-reliant. 

Pierre. Pierre had been in the U.S. for only two years and his English was 
difficult to understand. He had earned a D in the calculus course he had just completed so 
he signed up for individual counseling once a week but we did not meet until the end of 
the fourth week of class because of miscommunication. He reported no difficulties with 
mathematics in his early schooling. His anxiety scores were low and his Belief savvey 
results indicated a mastery-oriented approach to mathematics learning although his 
beliefs were somewhat more procedural than conceptual (2.5 on the I to 5 scale). 

We first met after Exam #1 where Pierre earned only a 68%. He put this down to 
having to take the exam early because of a prior obligation but his C^ on Exam #2 
seemed to point to something more. Pierre was in the B"/B* range on the conceptual 
multiple-choice and symbol section but in the D"/F^ range in the computational section. It 
did not seem that he had any fundamental problem with his arithmetic or algebra, 
although his operation sense (like Lee's), was inadequate (Appendix H, Table HI). He 



321 

seemed to have an over-inclusive approach to his learning. In his reported 17 hours per 
week of homework he surveyed and studied the greatest amount of material possible 
including extra material he asked Ann for and Pierre met with Ann and with me often. 

Because Pierre gathered and worked on so much, he was not mastering the 
mathematical computational material focused on in class, and he at times confused the 
extra material for material he was meant to use. In addition he approached the 
mathematical computation in a very procedural way, separate fi"om its conceptual base. 
For the third exam I suggested that he focus on the course material. When he did not and 
earned a D , I forceflilly confronted him before the fourth exam with the likelihood that if 
he did not change his approach he would get another D. He seemed a little shocked by 
my forthrightness but this time he listened. On Exam #4 Pierre earned a 91%, losing only 
one point on the computational section! When he came to tell me, he was very pleased 
and a little surprised at how much difference this strategy change had made. 

Pierre's English language difficulties contributed the most telling perspective on 
his performance in the classroom. It was clear that he had to use much of his energy to 
comprehend the material and to understand the organizational decisions. He did not 
collaborate with other students during problem-working sessions. Pierre did contribute a 
little in class (an average of once per lecture discussion) but his English continued to be a 
challenge for him and his peers. Although it dominated his class presence, it was not the 
main issue in his struggle to get a good grade; rather that issue was whether he was 
willing to give up his over-inclusive strategy to take a strategic approach. 

With much improved grades on Exam #5 and the MINITAB projects and a 
reasonable score on the optional comprehensive final to replace his lowest test grade. 



322 

Pierre went on to earn a B" in the course, much better than the D he was earning through 

the third exam. He retook Calculus I in the spring of 2001 and with this new approach 

earned a B^ to replace his original D. 

Jamie. Since Jamie is a focal student (see chapter 6) I will review her profile only 

briefly, chiefly to explain why I believe she falls in this category. As with Lee and Pierre, 

once we had ruled out arithmetic and algebra knowledge base issues as a central concern 

and began to reconnect Jamie to her secure mathematics base, counseling could focus on 

her central affective issues, which in Jamie's case was her severe anxiety as revealed in 

her Feelings survey, metaphor, and presence in class. Her shy personality had interacted 

with classroom teachers and family theories, and caused her to question her ability in 

mathematics Work on repairing damaged mathematics and mathematics teacher 

attachments, replacing her negative internalized teacher presences with positive ones, and 

supporting healthier interactions with the mathematics classroom personnel resulted in 

significant reduction in her anxiety, an improved sense of her mathematics self, and a B"^ 

in the course. However, her slightly more procedural than conceptual beliefs did not 

change and her performance orientation remained (see chapter 6 for a detailed account of 

Jamie's course of counseling). 

Students with Undermined Mathematics Self and Inflexible Mathematics 
Relationship Patterns 

Autumn and Mitch fell into this group. Like Lee, Jamie and Pierre, they had 

sound-enough mathematics preparation and compromised self-esteem emanating from an 

undermined mathematics self but unlike Lee, Jamie, and Pierre they did not seem willing 

to change their counterproductive ways of protecting their undermined mathematics 

selves. Their primary achievement motivation was for performance (certain grades) 



323 

rather than learning. They had achieved quite well in mathematics at times in the past but 
had also gotten disappointing results. They saw themselves as capable procedural 
mathematics students, but feared and resisted both problem-solving and the conceptual 
demands that were made on them. They did not want to risk exploring conceptual links. 
This approach resulted in a learned helpless orientation in conceptually demanding or 
problem-solving situations. Their underlying understanding of the algebraic variable was 
good-enough to support some conceptual exploration and their facility with arithmetical 
processes was adequate, although there was some question in my mind about operation 
sense. They tended to avoided open-ended questions (cf Autumn's efforts on Arithmetic 
for Statistics assessment, archived). Both had an overall negative attitude to themselves 
doing mathematics that could be classified as mild to moderate mathematics depression. 
They maintained detached distance from mathematics teachers and peers. 

These students seemed to have the most difficulty of all students in the class with 
any change of approach in how a class was taught and managed; their strong conservative 
impulse (cf Marris, 1974) led to strong resistance against change. It seemed that painflil 
or disappointing experiences with mathematics in the past had led to their building 
defensive barriers around their relatively sound but fearful mathematics selves to guard 
against scrutiny or further assault. They seemed inflexible and unwilling to give up their 
defensiveness in order to risk growth in understanding and achievement. 

Autumn. Autumn said she signed up for mathematics counseling to help me with 
my research. Although she reported disappointment with herself for not pursuing and 
succeeding in the algebra through calculus sequence, she was confident of success in 
PSYC/STAT 104 and did not want to explore conceptual connections or try to develop 



324 

her admittedly poor problem-solving abilities to become more mastery-oriented. She 
wanted a good grade rather than a conceptual understanding of the material and she 
maintained a performance motivation to learning statistics (see Table H3). She was a 
voluntary loner in class and maintained a detached distance from both the instructor and 
me. 

From her middle and high school mathematics history it became clear that 
Autumn's performance motivation had prompted her to take an easier class in order to 
earn an A. Her detached independence prevented her from getting the help she needed 
when she did try a harder class, particularly her advanced Algebra II class where she had 
a poor background because of the easier Algebra I class she had taken to get her A. Her 
low grade in advanced Algebra II had in turn contributed to her disappointment with 
herself, her compromised mathematics self-esteem, and mild to moderate mathematics 
depression that was evidenced in her Metaphor and responses on the JMK Affect Scales. 
Autumn's depression was not allayed by her consistently high grades in the course. 

If Autumn had participated in counseling designed to help her understand these 
connections and also supported her in exploring conceptual links and problem- solving, 
the current course experience might have developed her self-reliance and sense of 
mathematics self and perhaps even broken up her mathematics negativity. As it was, in 
counseling Autumn was willing to report her mathematics history, discuss her survey 
responses, and take the Algebra Test (a sound level 4) and Arithmetic for Statistics 
assessment (see Appendix H, Table HI), but she resisted doing exam analysis or 
exploring statistical procedures and concepts. Over the course, she remained relatively 
inflexible. Her procedural beliefs and learned helpless orientation changed little and her 



325 

abstraction anxiety score increased from 2.9 to 3.3 on the 1 to 5 scale although she 
reported that her confidence in her mathematics ability had improved (see Appendix H, 
Table H3). 

Mitch. Mitch signed up for mathematics counseling because he needed to erase an 
F from his GPA. That goal was admittedly limited but his self-reported rigidity and 
resistance to change jeopardized his achieving even such a limited goal. He did not want 
to explore his affective problems with mathematics although he alluded to them. If he had 
been willing to explore his metaphor of Inspector Javert"' as mathematics relentlessly 
chasing him through the years, he might have felt less beleaguered. Since he was not 
willing, what we did in the mathematics counseling was to work on the statistical 
problems at hand as I affirmed Mitch's sound mathematics self (e.g., his level 4 
understanding of the algebraic variable on the Algebra Test) and tried to help him 
reconnect with it. I helped him notice that not changing his approach from his failed 
attempt at the course was negatively impacting his attempt to do better this time. Through 
the third exam he used a formula sheet of the type his former teacher had allowed despite 
my pointing out this instructor's more generous criteria that allowed the inclusion of 
more information. His extreme negativity on the JMK Mathematics Affect Scales at the 
first session did abate somewhat but only two responses were on the positive end of the 
scale by his last session (6: mathematics achievement, and 7: making mathematical 
decisions). He made good-enough adjustments, earned a B to replace the F, and he is 
finally safe from Inspector Javert's pursuit; he never has to take another mathematics 
course at least as an undergraduate. 



326 

Category III Students with Underdeveloped Mathematics Selves: Mathematically 
Underprepared with Low Mathematics Self-Esteem 

The same number of students in the class had underdeveloped mathematics selves 

as had undermined mathematics selves. Those with underdeveloped selves fell into 

similar sub-types as those with undermined mathematics selves, that is, malleable and 

flexible/disorganized. 

Underdeveloped Mathematics Self Students with Malleable 
Relationship Patterns 

Karen, Mulder, Brad, and possibly Floyd were students in this study with 
underdeveloped mathematics selves who evidenced malleable mathematics relationship 
patterns. They had a history of struggling and/or not trying, poor mathematics 
achievement, and little (if any) feeling that they had ever understood. Like the adequately 
prepared students, they experienced relatively more success in other subjects. They were 
interested in understanding mathematics but felt capable of learning it only procedurally, 
if at all. They were more learning- than performance-motivated and were open to 
developing conceptual understanding once they believed they could, but all (particularly 
the men) seemed to fear risking the effort to understand, in case they found that they were 
incapable. 

Karen, Mulder, Brad and Floyd each had mathematics knowledge gaps evidenced 
in a low understanding of the algebraic variable and possibly also in arithmetical number 
and operation sense deficits. In their attempts to deal with the discomfort engendered by 
being in a setting where they felt lost and incompetent, these students had developed 
compensatory procedures and approaches that included avoidance, busy work, 
memorization techniques, under or overconfidence, external blame, and hostility. Each 



327 

had an underdeveloped and shaky mathematics self that produced negativity and empty 
depression on the one hand, or unrealistic bravado and resistance on the other. 

These underprepared, malleable students differed from the inflexible, adequately 
prepared students because they retained their learning motivation and an openness to 
learning conceptually despite their having experienced mathematical neglect."" In 
contrast, the inflexible adequately prepared students who also presented with 
mathematics depression, had experienced some mathematical success and had developed 
a good-enough knowledge base. Nevertheless they exhibited independent detachment and 
personal rigidity, performance motivation and resistance to problem-solving and 
conceptual learning. 

Karen. Since Karen is a focal student I briefly review her profile in terms of her 
mathematics preparedness and self characteristics. Karen's negativity about herself, the 
class, and mathematics, along with her hostile detachment relational pattern with teachers 
and peers and her knowledge base gaps were evident early. A picture of her moderate 
empty mathematics depression emerged as relating to an underdeveloped mathematics 
self (see chapter 2, Self Psychology). She was learning- rather than performance- 
motivated but took a procedural approach to mathematics because she did not believe she 
co?//£/ understand conceptually (although she wanted to). At the start of the course Karen 
consciously attributed bad outcomes to external sources, and in the counseling setting I 
had to overcome my countertransference reaction of feeling hopeless and depressed on 
her behalf I challenged her external control beliefs with evidence, provided mirroring of 
her tentative self, helped her see the instructor as on her side rather than against her, and 
offered enough structured guidance and course management that she was willing to 



328 

consider changing her external control to internal and to take responsibility for what she 
realistically could do. Her mathematics depression abated somewhat, she became more 
self-reliant, her grades improved, and her beliefs about mathematics became significantly 
more conceptual. Her overall confidence in her own ability to understand conceptually 
grew only slightly, however. Since the course was taught with manageable limits that she 
could handle with strategy and effort, what we did was good-enough (see chapter 6 for a 
detailed account of Karen's course of counseling). 

Mulder. Because Mulder is a focal student I highlight his characteristics briefly. 
Mulder had not really experienced success in mathematics, at least in high school. He 
"knew," based on his theory of family genes that he could succeed but he had not really 
tried. When he did try after Exam #1 in the class, he found that he could handle the 
mathematics computations but he struggled with the conceptual multiple-choice 
questions. Rather than mirroring his emerging prowess and supporting its application to 
the difficult multiple-choice, I was somewhat dismissive of that success. I pushed him on 
the multiple-choice and he resisted. 

It was not until we tried a counseling intervention suggested to me at my 
supervision session and I withdrew my counterproductive countertransference stance that 
he was able to overcome and succeed (see chapter 6 for a detailed account of Mulder's 
course of counseling). 

Brad. Although Brad's bravado was more extreme and more unrealistic than 
Mulder's, it seemed to have stemmed from a similar source — his underdeveloped 
mathematics self It brought forth a similar but more extreme countertransference 
reaction in me. I bristled at his we're-the-adults-here way of relating to Ann and me in 



329 

class. Unlike Mulder, Brad had tried PSYC/STAT 104 once and failed it, a fact that he 
seemed to almost inadvertently let slip in study group. He had, in contrast, written on his 
survey that he expected an A in this class and had earned a B in his last mathematics 
class. Algebra. He wrote on his metaphor survey that "anyone can do well" if he allows 
enough time and energy, yet he seemed ambivalent about doing that himself He was 
surrounded by women at work, and had a woman as his superior. His motivation for 
doing this class was to get a degree that would allow him to change to a more male- 
favored position. His conflict seemed to be around a fearful sense of not being capable of 
doing the mathematics, combined with a desperate need to be able to do it. He was taking 
a risk enrolling again, and my scolding and pushing him rather than supporting him in 
this effort was not helpfiil to him. Unlike Mulder he did not stand up to me but oscillated 
between avoidance and non-strategic effort in a way that did not achieve any more than 
marginal results. 

Floyd. The data I gathered on Floyd (from class surveys, the Statistics Reasoning 
Assessment and his Exam #1) revealed a similar bravado and resistance to getting the 
help he needed that Brad and to some extent Mulder exhibited. Like the other men in this 
group (it seemed) his grade hopes and expectations (both As) were unrealistically high, 
especially in light of his 42% failing grade on Exam #1 (see Appendix H, Table HI). He 
exuded confidence in class and declined the offer of mathematics counseling. However, 
like the other malleable underprepared students, Floyd's achievement motivation was 
more learning- than performance-oriented and his sound understandings on the Statistics 
Reasoning Assessment were the fourth highest in the class (10 of the 20) (see Appendix 
H, Table H2). 



330 

Analysis of Floyd's Exam #1 efforts"" revealed what appeared to be minimal if 

any prior study or practice, a somewhat surprising ignorance of basic statistical concepts 

such as median and mode (the only student in the class to show such ignorance), and 

probably a poorly constructed formula sheet. He did not make errors that indicated 

arithmetical gaps or misconceptions but there was too little data to assess that accurately 

or to assess his understanding of the algebraic variable. He overcame his resistance to 

getting help too late. He asked me for an appointment (just before Exam #2) but he did 

not come and then stopped attending the class. 

Underdeveloped Mathematics Self Students with Inflexible or Disorganized 
Mathematics Relational Patterns 

Kelly. Kelly had a history of poor mathematics achievement. She had deficits in 

number sense, operation sense, and understanding the algebraic variable. She had 

performance motivation and procedural beliefs, high levels of anxiety on all scales, and a 

learned helpless orientation to mathematics learning. Kelly had belief and anxiety scores 

similar to Karen's (a malleable student with an underdeveloped mathematics self) except 

that Karen was significantly more learning-motivated (3.5 compared with Kelly's 2.5). In 

addition, Kelly's externalized surprising-to-her "sudden storm" metaphor for 

mathematics, her and her mother's blaming her mathematics difficulties on something 

she felt was out of her control (a learning disability), and her relational pattern of 

dependence on both the instructor and me filled out a picture of her periodic sense of 

mathematical self disintegration. I allowed myself to be drawn into this vortex and was 

not able to help Kelly avoid another failing experience. My suggestions for counseling 

that may help such a student avert failure are discussed below. 



331 

Summary of Mathematics Self Categories 
From this analysis of characteristics, behaviors, and responses to counseling the 
three student categories emerged according to how students' mathematics selves had 
developed and what that implied about their present approaches to mathematics learning. 
These categories are summarized in Figure 7.1. 



Categories of Students According to Mathematics Self Development 

Category I Students with Sound Mathematics Selves: Mathematically Well- 
Prepared with Sound Mathematics Self-Esteem. Defined by soundness of arithmetical 
and algebraic knowledge base and absence of any experience of assault or questioning of 
mathematics ability or achievement, resuhing in a sound current mathematics self 

Type A: Sound Mathematics Self and Productive Relationship Patterns: 
e.g., Catherine and Robin 

Category II Students with Undermined Mathematics Selves: Mathematically 
Adequately Prepared but with Compromised Self-Esteem. Defined by sound-enough 
arithmetical and algebraic knowledge base and a variable experience of achievement with 
or without outside assauh on student's mathematics self concept, resulting in a relatively 
sound but undermined and vulnerable mathematics self 

Type A: Undermined Mathematics Self and Malleable Relating Patterns: 
e.g., Jamie, Lee, and Pierre 

Type B: Undermined Mathematics Self and Inflexible Relating Patterns: 
e.g.. Autumn and Mitch 

Types A and B are differentiated by their affect and relational patterns developed around 
vulnerable and ambivalent mathematics selves. 

Category III Students with Underdeveloped Mathematics Selves: Mathematically 
Underprepared with Low Self-Esteem. Defined by serious algebraic and/or arithmetic 
deficits or underdevelopment and a history of poor achievement resulting in an 
underdeveloped mathematics self 

Type A : Underdeveloped Mathematics Self and Malleable Relating 
Patterns: e.g., Karen, Mulder, Brad, and possibly Floyd 

Type B: Underdeveloped Mathematics Self and Inflexible or Unstable 
Relating Patterns: e.g., Kelly 



Figure 7.1. Mathematics self development categories of PSYC/STAT 104 participants. 

After I analyzed the student participants fi"om PSYC/STAT 104 into these 
categories, I was able to sort the counseling approaches I found to be appropriate and 



332 

helpful against these categories. I found that approaches indicated were closely related to 

the categories and also to students' affected dimension of mathematics relationality. 

Integrating Relational Mathematics Counseling 
with Mathematics Tutoring: An Analysis 

Dimensions of participants' mathematics relationality were interdependent, but 
some students had more pronounced difficulties in one dimension than the others. The 
categories of participants' mathematics ftinctioning identified in this chapter (see Figure 
7. 1) seemed to be related to the problematic dimension (particularly the self dimension) 
and the depth and type of the relational difficulty. 

As I have demonstrated, mathematics knowledge base deficits interacted 
predictably with students' mathematics self development and it was these two factors and 
their interaction that pinpoint a student's profile type. Past negative teacher-student 
experiences formed internalized presences that interfered in the present and had caused 
damage to mathematics selves. These negative teacher-student experiences had also 
caused damage to mathematics and mathematics teacher attachments (cf Jamie). 

Difficulties with establishing or maintaining secure attachments to mathematics 
and/or mathematics teachers also strongly affected their present relational patterns and 
mathematics functioning (cf Jamie and Karen). Mathematical and counseling 
instruments and techniques for diagnosing and treating difficulties in one dimension at 
times resulted in improvements in another; in other cases they proved inappropriate and 
even counterproductive in dealing with another dimension. 

In the following three sections I present my analysis of mathematics relational 
counseling for each of the three relational dimensions that Mitchell (1998) identified, 
which form the basis for my approach. In this analysis I show how a student's category of 



333 

mathematics self identified above affected the counseling needed and its possible 
outcomes. 

Mathematics Counseling and the First Dimension: Self 
Some students with pronounced defects in their mathematics selves presented as 
either unrealistically negative (underconfident) or positive (overconfident; in either case, 
unrealistic) about their mathematics self (Category III, Type A students, cf Karen, 
Mulder). Others clung to the counselor with little or no sense of having an independent 
mathematical existence (the Category III, Type B student, Kelly). Students whose 
mathematics self was relatively sound but had been undermined so that they were no 
longer confident in it were likely to present with inappropriately severe anxiety (Category 
n. Type A women, cf Jamie, Lee) or with a rigid resistance to change or risk (Category 
n, Type B students, cf Autumn, Mitch) depending on their attachment patterns. I was 
able to use the following means to explore disordered self relational patterns: 

1 . Investigation of the mathematics knowledge base: using diagnostic assessments of 
arithmetic and algebra, class exams, and/or learning modality and style checklists, 
and 

2. Investigation of self relational patterns by: 

(a) Investigation of pronounced ongoing negativity/depression on the JMK 
Mathematics Affect Scales, Learned Helpless-Mastery Oriented Scale, 
generalized negativity, and underestimate of the mathematics self 
(underconfident). Clues lie m Metaphor, in self-statements in counseling, and 
in my countertransference feeling of depression or despair for the student's 
prognosis (cf Karen); 



334 

(b) Investigation of pronounced discrepancies between a student's elevated 
perception of his mathematics self when compared with a realistic assessment 
of mathematics self (overconfident). Clues lie in Metaphor, in self statements 
in counseling, and in my countertransference of first believing and then 
wanting to dispute inflated and unrealistic self assessment and to deflate it (cf 
Brad, Mulder); 

(c) Investigation of anxieties that seemed disproportionate with measured 
levels of mathematics competency on the algebra and arithmetic assessments 
and/or exams. Clues lay in Feelings survey scores [and possibly Metaphor] 
relative to sound mathematics diagnostic scores [and possibly class exam 
scores] (cf Jamie); 

(d) Investigation of inappropriate dependence on counselor combined with 
lack of focus, willingness, or belief in ability to engage cognitively in the 
mathematics. Clues lie in Metaphor, in self statements in counseling, and in 
my countertransference feeling of being sucked into a bottomless pit (cf 
Kelly); 

(e) Investigation of a marked discrepancy between personas in different 
settings, for example, in class compared with the counseling setting (cf 
Robin). 

Students with moderately to severely underdeveloped mathematics selves were 
underprepared mathematically and their self-esteem was consequently low. They had 
inadequate scores on the algebra diagnostic (and some also on the arithmetic diagnostic) 
and low scores on the first exam in the course. Where I treated these problems 



335 

effectively, I mirrored sound mathematics thinking and course strategy practices so 
students' sense of their own competence would become both realistic and hopeful. I 
provided myself as a good-enough mathematics parent image for students to idealize and 
model themselves on but I subsequently provided manageable finstrations and 
disappointments so they could withdraw dependence and grow into their own 
competence. I found this easier to accomplish with Karen, who presented with symptoms 
described in 2. (a) above. 

I did not deal as well with the students described in 2. (b) Brad, or 2. (d) Kelly 
because I did not discern soon enough that their root problem also lay with their 
underdeveloped and vulnerable mathematics selves. With Mulder and Brad, for example, 
instead of mirroring areas of real competence I tended to act out my countertransference 
reaction to deflate their overly positive opinions of themselves. My inappropriate 
approach tended to increase their overt grandiosity and their resistance to or avoidance of 
the task but with Mulder, supervision advice and my becoming aware in time of the part I 
played in his resistance, a positive outcome was achieved. With Kelly, I was drawn into 
her vortex and tried frantically to give her all she thought she needed instead of mirroring 
her evidenced competencies and providing bounds she could not establish for herself 

Category II students who had suffered some short-term and/or long-term blows to 
their mathematics selves did have adequate underlying mathematics selves (and 
knowledge bases) but they had been undermined. They needed not so much to develop 
their mathematics selves through mirroring and permission to idealize, but needed rather 
the offer of a secure base and help with repairing damaged attachments (i.e., techniques 
of the interpersonal attachment dimension, see below). 



336 

Mathematics Counseling and the Second Dimension: Internalized Object 
(Internalized Presences from the Past). 

All students' behaviors and expectations are influenced by their prior mathematics 

learning experiences. But students suffering from the undue negative influence of their 

internalized mathematical presences (internalized objects) typically behaved towards the 

instructor or peers or tutor in ways that were incongruent with present realities. In this 

study Jamie was most affected this way, but it does not appear that a problem in this 

dimension is restricted to or indicative of a particular student category. I found that 

certain learning style differences and learning modality preferences could be confounded 

with a problem with this dimension, however, so assessment needs to be careful. The 

most important diagnostic data came from: 

1. Investigation of the mathematics knowledge base: using diagnostic assessments of 
arithmetic and algebra, class exams, and/or learning modality and style checklists, 
and 

2 . Investigation of internalized mathematics relational patterns by : 

Investigation of obser\'ed behaviors in class or study group or counseling that 
seem incongruent with the way class members related to each other, the 
teacher or the counselor/tutor. Clues lie in Metaphor and History Profile; in 
my sense of student's transference that was very different from reality [e.g., 
dangerous to Jamie]; in my countertransference feelings that I should act 
differently from what I believed would be appropriate [e.g., stay away and not 
ask questions so as not to cause damage]. 
I found that discussion of metaphor and mathematics history quickly uncovered 
bad internalized teacher presences that interacted with personality and caused present 



337 

mathematics teachers to seem dangerous, for example. Counseling involved support in 

close analysis of the instructor and the mathematics counselor to see if they could 

displace the bad object (presence), and devising relational assignments (cf Jamie's 

assignment to ask Ann a question and make an appointment with me) based on a new 

more realistic evaluation. If a student's life were constrained by extremely bad 

internalized mathematics presences, however, such a straightforward process would 

likely not be possible. In that case, the mathematics counselor should not proceed except 

as a team member with a mental health counselor. 

Mathematics Counseling and the Third Dimension: 
Interpersonal A ttachments 

Students in the study who had developed insecure attachment patterns with 

teachers presented as avoidant, overly dependent, ambivalent, or fearfiil of the teacher or 

counselor. Certain personality styles seemed to be conflated with this dimension, 

however, so diagnosis has to be carefiil. Some students suffered from an insecure 

attachment to the mathematics, they presented most often as procedural in their 

mathematics (cf Autumn), with associated uncertainty about their ability to do 

mathematics, and separation anxiety in exams. The most important diagnostic data for 

insecure attachment to teacher or mathematics came from the following: 

1 . Investigation of the mathematics knowledge base using diagnostic assessments of 
arithmetic and algebra, class exams, and 

2 . Investigation of interpersonal mathematics relational patterns by : 

(a) Investigation of observed m>oidant or dependent attachment behaviors in 
class or study group or counseling. Clues lie in Metaphor and History Profile; 
in my sense that a student's transference was keeping me at a distance 



338 

personally (cf. Karen), or that she was excessively needy for my presence; or 
that she was ambivalent; in my countertransference that I should try to gain 
her approval or that I wanted to escape, or that I was confused and continually 
moving between the two reactions; 

(b) Investigation of a student's apparently mrwarranted insecurity in her 
ability to do the mathematics at hand. Clues lie in responses to the Beliefs 
Survey and Procedural/Conceptual and Learned Helplessness sub-scales; 
history of intermittent success and relative failure in mathematics; more 
anxiety on tests than seems appropriate given preparation and mastery of the 
material. This lack of a secure base in the mathematics seemed to be the result 
of a history of procedural transmission teaching and never having truly 
understood the mathematics or a history of having been suddenly separated 
from a secure mathematics base. 
When students evidenced an insecure teacher attachment pattern, my counseling 
role was to provide a secure teacher base where they felt mathematically accepted and 
safe so that they could begin to explore on their own and risk taking paths that might be 
wrong, so as to eventually become self reliant. I also needed to help the participant 
reevaluate the present instructor and her approaches and begin to receive rather than 
reject her good offerings (cf Karen""). With students showing an insecure mathematics 
attachment pattern, my counseling role was to help them rediscover the existing sound 
basis in mathematics from which they had been separated (cf Category II students, 
Jamie, Mitch). These students' mathematics separation anxiety dissipated as their security 
in the mathematics grew. When I helped insecure procedural learners link their 



339 

procedures with the underlying concepts they began to estabUsh their own secure 
mathematics base. 

Summary of Brief Relational Counseling Analysis 

I have analyzed here the application of brief relational mathematics counseling 
according to the dimension of student relationality: self internalized presences, and 
interpersonal attachments. I have also shown the interactions among the categories of 
students according to mathematics self development, which I identified earlier in the 
chapter, and the counseling approaches that are applicable. It is not possible to establish 
a causal link between student outcomes and the counseling because of the many variables 
at play. However, in the process of the counseling, the participants and I did identify 
relational conflicts by attending to patterns of relational episodes and we attempted to 
resolve them. As could be expected counseling Category II students required less 
emphasis on the mathematics itself than for Category III students because their greater 
level of mathematics preparation enabled them to proceed without as much mathematical 
support once they were reassured of their competence. Category III students needed more 
mathematical support throughout and each category of participants benefited from 
relational counseling to help them resolve the relational conflicts that had arisen over 
their learning histories. In most cases, counseling worked well enough that focal 
participants and others felt their originally questionable course prognosis changed and 
they succeeded. 

In chapter 8 I will reflect on my relational approach and its components and 
suggest directions for further research. 



340 



' Because this is an odd numbered chapter, I use "he," "him," and "liis" for the generic third person. 

" Catherine reported tliat she Hked matliematics and was confident in her ability to do well. She was more 
conceptually than procedurally oriented towards mathematics learning (3.5 on a scale of I procedural 
through 5 conceptual, on tlie pet-Mathematics Beliefs Survey and 3.8 the highest in tlie class on the post- 
survey). Her presence in the class did not cliange over the course. She seemed comfortable; she was quietly 
(contributing only an average of .27 responses or questions per session) confident during the lectines, 
appearing to be processing and understanding tlie material; she worked on her own during problem- 
working sessions but was willing to interact witli a neighbor if Uic neiglibor initiated it (e.g., Mulder, Class 
7). Since Catlierine, unlike Lee, did not seek conceptual links during lecture discussions or problem 
working sessions nor participate in study groups or individual counseling, she may have been using some 
of her 5 hours per week of homework time doing that. Her mathematics testing anxiety was initially the 
lowest in the class (1.5 on a scale of 1 [none] through 5 [extreme]) but uicreased considerably to 2.1, 
nevertlieless remaining relatively low. 

■^ RN to BSN RN = Registered Nurse; BSN= Bachelor of Science in Nursmg. 

" She had the characteristics of Davidson's (1983) Mathematics Learning Style II learner or Krutetskii's 
visual-pictorial (see chapter 2). 

*" Evidenced in their Level 4 understanding of the algebraic variable on \hc Algebra Test hxA somewhat 
variable Arithmetic for Statistics responses. 

" From Alain Boublil and Claude-Michel Schdnberg's musical adaptation of Victor Hugo's novel Les 
Miserables. 

™ Karen did have a detached pattern of relating but it was defensive rather tlian independent and her 
mathematics depression was more prominent. All Category II, Type A students had issues of control but for 
Category II, Type B students it was much more prominent. 

™ For only four of fourteen questions did Floyd apjjeared to understand what the question was asking for, 
know the formula, use the formula correctly and achieve a realistic answer; for two questions he used 
appropriate formulae but did not seem to understand the meaning of the question so substituted incorrect 
(though not unrelated) elements and achieved unreahstic answers; on six questions he did not seem to 
understand the meaning of the question; and on the other two he showed only partial understanding of the 
question and of the procedure required achieving a somewhat realistic answer in one of them. 

" I accepted her anger and her depression and offered mathematics counseling as a secure base from which 
she could do the course. Towards the end of the course she had developed more self-reliance and chose 
when she did and didn't need to come for help. She also clianged in her stance towards the instructor 
learning to appreciate and rely on her. See chapter 6 for fiirther discussion. 



341 

CHAPTER VIII 

EVALUATION OF THE PILOT STUDY AND RECOMMENDATIONS 
FOR FURTHER RESEARCH 

When Jamie and I first met, mathematics was a storm that she was afraid would 
come back so she stayed inside. By the end of the counseling, it had changed to a partly 
sunny day and she could go out with her umbrella. Jamie's relationship with mathematics 
changed as she went to class, met with me in counseling, and struggled to understand and 
resolve the central conflict that had been sabotaging her conscious desire and her ability 
to do well in mathematics. Through this study I have developed a new way of providing 
mathematics support over a college semester, one that incorporates relational and 
cognitive counseling approaches. Here I turn from looking at the particulars of each 
participant's experience of counseling to looking at what those particulars might tell 
about the counseling approach itself as it emerged in this pilot application. 

People can change. I found that out in this study. By crossing the lines drawn in 
traditional mathematics support in order to incorporate a relational counseling approach, I 
first changed how I looked at students and at myself, and they in turn changed how they 
looked at themselves, at the instructor and me, and at mathematics. We found that we 
could disembed ourselves from our entrenched theories about ourselves and each other 
and change our counterproductive patterns of relationship in mathematics learning when 
we recognized, explored, and challenged those patterns. To do so we each had to cross 
traditional lines to widen appropriate and useful objects of attention in academic support 
settings. We had to consciously attend to our relationships. 

To explain fiorther, I first evaluate this process of counseling and its elements, and 
evaluate the student categorizmg system that emerged. Next I assess the limits of the 



342 

approach as it developed. Finally I discuss the limitations of the study and suggest future 
directions for research and development of this approach. 

Understanding the Student: Who is She' and How Do I Know? 
How the Counselor Role Changed What I Knew 

As an educator I used tests and surveys to try to classify each student's 
mathematics cognition and affect; I took what she told me about herself at surface value. 
As a counselor I gradually learned to listen not just to her "I" statements but to her 
behaviors, her metaphors, her "she should" and "they are" and "it is" statements, her 
transference (the role participants seemed to be casting me into), and to my 
countertransference (my feeling constrained to act out or react against the role imposed 
on me by the participant's transference). I looked for links between her history and her 
current mathematics performance. Jointly we looked at her exams and reviewed her 
grade, her thinking, her feelings and beliefs, her effort, and her contradictions. The chief 
difficulties 1 faced in understanding the student were (a) the now-dynamic nature of our 
relationship (The student changed, I changed, and our interaction changed.) and (b) the 
reality and power of the unconscious: The student spoke honestly about her realities, yet 
there was often good evidence that seemingly contradicted what she said; sometimes she 
reacted in surprising ways that seemed incongruent with the present reality. These 
difficulties provided the richest sources for understanding her (and me) as we worked 
together. 

When I heard or found or sensed contradictions, at first I was angry and mentally 
accused the student of falsehood or cowardice (i.e., Brad's assertion that he earned a B in 
his last mathematics class when he had actually failed his last class). At times I felt hurt 
(i.e., Jamie only hinting and only at the end of the class that she was repeating 



343 

PSYC/STAT 104). I chose sides and went with one assertion, dismissing or refuting the 
other side." But when I drew on relational counseling insights through supervision and 
further readings, I recognized these contradictions and the conflict that they created to be 
equally genuine realities for the student. It was that very conflict that needed to be 
brought to consciousness and resolved. Formerly, in my educator-only role, I was not 
aware of wrestling with such nagging contradictions so I had not brought them to the 
student's attention. They remained as the "elephant in the living room" — ^known about by 
both of us at some level but unacknowledged. This lack of awareness or resolution of a 
student's conflicts may have precluded the possibility of unproved mathematics mental 
health and success in mathematics courses. 

When I encountered what seemed to be willful refusal to allow me to see areas of 
vulnerability or to help change behaviors or approaches (particularly with Autumn and 
Mitch), I felt frustrated. / could see what their problem was — why would they not discuss 
it with me, explore it, and resolve it? Through supervision some of my blind spots were 
identified. I had difficulty allowing a student to choose her own path, especially when 
that path seemed counterproductive to me. Instead I tried to get students to see their 
difficulties as I saw them and to change. In the cases of Autumn and Mitch, my behavior 
probably contributed to their becoming more entrenched in what I saw as their 
counterproductive approaches. 

Working through a relational conflict perspective allowed me to understand that 
students had developed their current patterns of relationships in their attempts to protect 
and defend their vuhierable mathematics selves. When I brought together my educator 
and counselor roles I began to understand how a student's mathematics history might 
have influenced her current ways of functioning within her overall relationality. This 



344 

integration yielded the three broad categories of mathematics student (each with at least 
one subcategory) that I described in chapter 7 (see Figure 7. 1 ). 

Developing Categories to Understand Students. 
Why Categorize Students? 

In my endeavor to effectively support the whole person doing mathematics, I 
needed to understand the range of variations in students' responses to their mathematics 
learning histories. The significance of these variations and similarities helped me notice 
details as part of a whole rather than being distracted by them. These variations also 
helped me understand that quite different- looking symptoms could stem from similar 
sources and might call for a similar counseling approaches (i.e., Karen's under- 
confidence and empty depression and Mulder and Brad's over-confidence and 
grandiosity both were expressions of underdeveloped mathematics selves that stemmed 
from mathematics underpreparation and low self-esteem). 
Emergent Categories 

I developed categorical descriptions that were determined by interactions between 
a student's history and adequacy of mathematics preparation, and her mathematics self- 
esteem. My analysis of the case data gathered in this study led me to suggest that the 
condition of a student's underlying mathematics self may be classified into one of three 
categories: (a) Category I: A sound flinctioning mathematics self; (b) Category II: A 
relatively sound but undermined and vulnerable mathematics self; and (c) Category III: 
An underdeveloped mathematics self. Mathematics preparation and related self-esteem 
were the principal discriminator of these three categories (see chapter 7, Tables 7.1 and 
7.2). 



345 

With my course participants, categories were further refined according to how a 
student had handled her compromised self-esteem, that is, by developing either malleable 
or inflexible (or unstable) mathematics relational patterns (see chapter 7, Table 7.3). 
These categories are not exhaustive, however. Conceivably if these distinctions are 
applied to other groups of mathematics students, other subcategories could be identified 
within one or all of the three broad categories. I suspect that these three broad categories 
are sufficiently explanatory to encompass all students. It is possible however, that 
Category III might be helpfully divided into two categories according to whether the 
student had a low level of the algebraic variable with adequate arithmetic skills or 
inadequate levels of both. Further support for my categorization scheme is shown in the 
fact that students from my preliminary research on practice are relatively easily 
categorized with this scheme (e.g., Mary as Category II, Type A and Jane and Cara in 
Category III, Type A, see Knowles, 1998, 2001) 
Comparison with Other Schemes 

The only similar attempt to classify mathematics students is the tier sort Sheila 
Tobias proposes (personal communication, March 16, 2001; see chapter 4, pp. 131-134). 
Tobias' &st, second, and third ("utilitarian") tiers more or less correspond to my 
Category I, Category II: Type A, and Category II: Type B, respectively. Her 
"underprepared" fourth tier and "unlikelies" fifth tier do not comfortably parallel any 
categories I found but my Category III: Types A and B have "underprepared" and 
"unlikely" characteristics. Tobias has researched her second tier in science classes 
(personal communication, April 5, 2003; Tobias, 1990 and her other tier categories come 
from experience and observation, again primarily based on science students although she 
applies them to mathematics students. The major contrast in our schemes is that mine is 



346 

based on underlying cognitive and relational differences in mathematics self development 
and learning history, while Tobias' focuses more on current cognitive preferences and 
behaviors. I find that mine provides more direction for counseling support and 
intervention. 

Evaluating Cognitive Categorizing and Counseling Instruments 

The principal means I used in this study to gauge students' mathematics cognitive 
functioning levels were: course assessments, the Algebra Test, the Arithmetic for 
Statistics assessment, and the Statistics Reasoning Assessment (SRA) (see Appendix C). 
All but the last provided valuable data both for categorizing students in order to develop a 
mathematics tutoring focus as well as for relational mathematics counseling. Taken 
together the first three data sources helped me sort students into the three categories of 
mathematics preparation: well prepared, adequately prepared, and underprepared (see 
also chapter 7, Table 7.2). A most important finding of this analysis for this group of 
students is that course the first exam grades did not provide in themselves an accurate 
indication of membership in a category or subtype of a category except perhaps for the 
students in Category I and to some extent Category III. The Algebra Test was a better 
indicator for the students who took it. It distmguished between students most 
appropriately described by Categories II and III. Students' arithmetic levels (as gauged 
by the Arithmetic for Statistics assessment and/or arithmetic samples gathered from 
exams, counseling and in class) discriminated well among all three categories. 
The Algebra Test (Brown, Hart, & Kuchemann, 1985; Sokolowski, 1997) 

I found the Algebra Test (see Appendix C) useful in mathematics counseling with 
students who scored at high concept levels of the variable but who had also developed 
negative or ambivalent beliefs about their own mathematical ability — Category II 



347 

students.'" A high level provided some proof that their negative views were not accurate, 
and this was more objective than my reassurances or even the evidence of their 
coursework. It was worthwhile to use some mathematics counseling time to take this test 
because their difficulties did not seem to be fundamentally mathematical. 

Once I established, for instance, that Jamie's level of the algebraic variable was 
high and not an issue for counseling, I determined to use this good result to refute her 
negative beliefs about her mathematics ability. The other three participants to whom I 
administered the Algebra Test during the course were all at Level 4 (see Appendix H, 
Table HI) and in each case this good result was used in counseling to allay concerns 
about each one's mathematical ability. Because Category II and III students' first exam 
results discriminated their category relatively poorly, the Algebra Test seemed to provide 
a more accurate way to clarify early her level of cognitive preparation especially when 
taken with Exam #1 grades and arithmetic preparation (see chapter 7, Table 7.2 and 
discussion). 
Arithmetic for Statistics (AFS) Assessment 

Each of the five students at level 4 or above on the Algebra Test took the 
Arithmetic for Statistics (AFS) assessment (see Appendix C) and performed adequately 
on it on at least seven of the eight categories tested.'^ The two students who were at level 
2 on the Algebra Test (see chapter 6, Table 6.1). Karen and Mulder performed quite 
differently on the Arithmetic for Statistics assessment. Mulder performed adequately on 
all categories except large integer number sense and statistical sense. Karen however did 
not perform adequately on any category and her performance on operation sense, place 
value/decimal sense for numbers of magnitude greater than 5, and open ended 
arithmetical thinking/problem-solving was inadequate (see chapter 6, Table 6.2). 



348 

I developed the AFS assessment as the course was proceeding in order to more 
clearly isolate participants' arithmetical conceptual and procedural difficulties related to 
the mathematical requirements of the course (see Appendix C). Despite its limitations^ it 
revealed more precise data about participants' arithmetical issues than I could observe 
anecdotally in class or counseling. With modification, I believe it should be administered 
early in the counseling process, so that arithmetical issues may be addressed more 
systematically with the arithmetically weaker students. Adjusted to satisfy issues raised in 
endnote v, it should be a useful tool to be added to the Algebra Test and used at the 
beginning of the counseling process. This would help students with specific weaknesses 
or problem areas that impact their confidence and progress in mathematics. 
Statistics Reasoning Assessment (Garfield, 1998) 

The SRA was not useful in category placement or diagnosis for the strategic 

mathematics counseling of students taking PSYC/STAT 104 (see Appendix H, Table 

H2). Changes in scores fi-om pretest to posttest did not parallel other changes students 

made over the course. This was not surprising because the course's design, direction, and 

implementation were not focused on confronting and changing individual students' 

misconceptions about statistics or probability, which the SRA was designed to measure. 

The primary focus of my counseling was to support students in their coursework so much 

of what is assessed by SRA did not match. 

Evaluating Affective and Relational Categorizing and 
Counseling Instruments 

College Learning Metaphor Survey 

Metaphor writing and analysis quickly provided rich, deep material that was 

directly relevant for both the participant and for me; it was key in establishing the central 



349 

conflict and relational focus (and in some cases the mathematics focus) for a participant's 
counseling. Given the brevity of available time the quick collection of data that revealed 
underlying issues was important. All but one participant found no difficulty in creating a 
metaphor and nearly all were open to jomtly interpreting and exploring the meaning of 
personal metaphors. 

The chief limitation in using metaphors lay in my tendency to assume that I 
understood when I should have remained open and probed more. It was easy to be 
diverted by other data and in some cases I initially failed to use those data in conjunction 
with the metaphor in order to see a clear common focus. In order to disembed the student 
from her own metaphor, both the student and I probed its meaning; explored its links to 
current practices, reflections, and automatic thoughts; explored ways to change; and 
finally, the student created a new metaphor to reflect on changes made. 

The shared analysis of the meaning of students' metaphors and what I learned of 
their deeper meaning to the student often provided a unifier or common thread and even 
provided vital missing clues to the relational conflict, the mathematical focus or both. I 
discussed these insights with some participants, and we explored the implications 
together. However, in these cases I initially understood only part of the meaning; as 
counseling progressed more data emerged from the metaphor in the context of the 
student's approach to the counseling, to the course, and to the mathematics.^' 

With some participants I found the initial link between the metaphor and other 
presenting data was less accessible to me. Thus I found a conscious formulation of the 
central conflict and dynamic foci more difficult. For example, along with persistent 
negativity on the JMK Mathematics Affect Scales that contrasted with their course 
performance, the metaphors of Category III: Type A woman and the two Category II: 



350 

Type B students were an important piece in the diagnosis of mild to moderate 
mathematics depression, despite behaviors that initially indicated otherwise. The 
Category III: Type A men's metaphors at first seemed active and positive but in light of 
these students' initial poor performance, somewhat grandiose. However, further analysis 
revealed that these men's metaphors indicated a sense of being outside of the 
mathematics; their metaphorical characters used elusive and discoimected clues to try to 
understand the alien or mysterious mathematics, and I saw that the metaphors truly 
provided an accurate representation of how the students viewed and approached 
mathematics.^" 

The Category III: Type B student's metaphor held rich though indirect material 
and early indicated a lack of realism on her part about how she might need to change in 
order to succeed in the course. 

Only one student refused to engage in exploring the meaning of a metaphor that 
seemed to me directly linked with his problems with the course (a Category II: Type B 
student). Even so, I was able to use the insights I gained to provide interventions such as 
giving him the Algebra Test to reassure him that he had the ability to opt out of his 
metaphor by passing PSYC/STAT 104. 
JMK Mathematics Affect Scales 

Both content and structure of this instrument made it extremely useful in the 
counseling situation (see Appendix B). The scalar design allowed for open-ended 
responses and its repeated use proved invaluable. The range from positive to negative 
allowed students to see their changes over time. Our shared discussion linked these 
changes to changes in their life circumstances, personal decisions, automatic thoughts, 
and unconscious patterns. Scale topics focused on students' immediate sense of their 



351 

mathematics self, world, and future (cf. Beck, 1977). The selected topics proved to be 
important but I found that they were difficult to address verbally at each session. My 
asking direct questions might have been perceived as accusatory or confrontational, and 
asking them at each session might have seemed to be nagging. The use of these scales 
avoided that conflictual situation. 

People who are negative about themselves, their world, and their futures often rate 
themselves more negatively compared with their peers than may be warranted. To 
measure this I would add a new item to the JMK scales to investigate this perception: 
Compared with others in this class, I do mathematics better than/as well as/worse than 
most of them (see Appendix B for the original and revised versions). 

There seemed to be a relationship between some students' metaphors and their 
responses to the JMK Scales. When a student's metaphor was negative, stable, and either 
passive (e.g., "cloudy" or "overcast") or indicating persecution (cf Inspector Javert) there 
seemed to be an underlyuig mathematics depression as measured on the JMK Scales, yet 
when a student's metaphor was negative but unpredictable (e.g., storm), mathematics 
depression did not seem to be generally present — anxiety seemed to be more of an issue. 
Beliefs Survey 

I found each scale: Procedural vs. Conceptual; Toxic vs. Healthy; and Learned 
Helpless vs. Mastery Oriented, taken with other data, to be especially relevant for 
different participants. The first scale differentiated Category I students from the others 
and discriminated somewhat between Types A and B in Categories II and III. In most 
cases when a scale was highlighted with a participant in counseling, she became more 
aware of its relationship to her approach to the mathematics. She was usually able to 
clarify how it was manifested in mathematics testing and learning situations. From this 



352 

she often became more able to change in positive directions. If a post-course meeting to 
discuss post-scores, changes, and current beliefs had been possible with each participant, 
developing and discussing a long-term plan for each participant's mathematics future 
might have been feasible 

A cluster of questions surveying learning versus performance motivation emerged 
in the post analysis as a discriminator between Types A and B in both Category II and III 
students, with Type A students being more learning-motivated than Type B students. A 
revised short Beliefs survey that highlights this factor is presented in Appendix C {My 
Mathematics Orientation). 
Feelings Survey 

Each of five students (except Lee) who reported very high mathematics testing 
anxiety"" signed up for mathematics counseling during the pretesting session at the 
second class of the course when I offered coimseling to all. Lee initially refused 
counseling but contacted me just before the first exam requesting support, citing her 
mathematics anxiety. These students also had the highest abstraction anxiety scores in 
the class (from 3.2 through 4.2 on a 1 through 5 scale). Jamie was the only one of these 
five for whom we eventually established the primary relational focus to be anxiety and 
the only one whose anxieties on this instrument all decreased substantially. I found this 
instrument to be useful in conjunction with other instruments in establishing a diagnosis 
although it did not seem precise when used alone (see Appendix C for the My 
Mathematics Feelings survey). 

Mathematics testing anxiety of the class increased slightly overall but the class' 
average responses to individual items are of even more interest. "Signing up for a math 
course" or "Walking into a math class" now evoked considerably more anxiety than at 



353 

the beginning of the course (from 2.6 to 3.1 and from 2.1 to 2.7 respectively). In contrast, 
"Waiting to have a test returned" and "Receiving your final math grade in the mail" now 
created considerably less anxiety than at the beginning of the course (3 to 2.6 and 3.1 to 
2.4 respectively). When taken with other evaluative data, these responses seem to 
indicate that in the context of this class students' anxiety levels had decreased as their 
control and achievement had increased but that this improvement did not generalize to 
future or other mathematics classes. In fact, the prospect or memory of other courses now 
evoked more anxiety. 

I paid little attention to the number anxiety results during the study since all 
number anxiety mean responses were at or below the mid point (3) of the scale and thus 
seemed to indicate low to moderate anxiety, especially when taken m contrast to the 
reported testing and abstraction anxieties that went as high as 4. 1 and 4.2 respectively on 
the pretests. I realize now that the two participants whom I early recognized to have poor 
number and operation sense (Kelly and Karen) had the highest number anxiety scores in 
the class at 3 and 2.9 respectively. Lee, who was considerably more competent 
arithmetically than they, had a relatively high score of 2.8. On the other hand, on the 
Arithmetic for Statistics assessment, although Lee was generally adequate, she 
nevertheless had a marginal operation sense that likely contributed to heightened number 
anxiety. I would now flag scores in the middle of the number anxiety scale for immediate 
investigation of a student's number and operation sense. 

For some individuals the changes in their feelings on survey responses confirmed 
the direction and efficacy of the mathematics counseling. For example, Jamie's testing 
and abstraction anxiety showed an overall significant decrease, with some aspects 
increasing while others decreased (see Appendix H, Table H3). In chapter 6, 1 discussed 



354 

the course of counseling with Jamie in detail. Her dynamic focus was specifically social 
anxiety'" that was intensified in the mathematics learning environment, and it was this 
and her related practices that we worked together to change. All aspects of Jamie's 
testing anxiety that had a self-focused social public component ("Walking into a math 
class," "Raising your hand in a math class to ask a question,"" and "Waiting to have a 
math test returned.") decreased over time. The aspects of Jamie's abstraction anxiety that 
had a self-focused social public aspect also decreased. By contrast, Jamie's anxiety about 
taking a final math exam in class increased. This seemed to have a mathematics self- 
competence, performance focus for her rather than a social self-focus. 

The Role as Relational Counselor Transform My Tutoring Work 
Counseling Use of Transference and Countertransference 

The new need to attend to transference and countertransference immediately gave 
me conscious access to a fund of analyzable and usable data that I had previously largely 
ignored in mathematics tutoring practice. In typical tutoring situations, transference 
usually remains unplicit as both student and tutor often continue to act out old patterns of 
interaction without the conscious reflection that my new approach encourages. 
Relationship patterns based on the student's internalized teacher presences of the past 
may pull or push the tutor mto assuming the teacher role they demand. She may on the 
other hand react against assuming roles she believes are toxic for the student's 
mathematics mental health. Because these relationship patterns are not brought into the 
open the student may resent the refiisal of the tutor to take on the expected role and the 
student's expectations are not realigned. But my new approach to the participants in this 
study allowed for this material to be brought into the open and dealt with consciously so 
that we could each adjust to more productive ways of relating. 



355 

In the academic setting, my use of insights gained from transference and 
countertransference was necessarily quite different from a psychoanalyst's use. Since the 
focus was on the mathematics and not on the resolution of personal psychological 
problems, interpretation and specific working through of transference was not 
appropriate. What was appropriate was noticing it and checking with the participant 
about shared insights. Most important was looking at patterns of interactions over time 
including the transference and countertransference so that a central relational conflict 
could be identified. 

I found it challenging to attend to the student's transference and to my 
countertransference. In the past, I had found myself on occasion acting in ways that 
surprised and concerned me — for example, believing whiners and joining them, almost 
doing a student's work for her, agreeing to work with a student much more than 
appropriate, scolding, or panicking with them. It had not at all been my practice in the 
mathematics tutoring situation to consider what these behaviors might be telling me about 
the students' history, personality, approach, and practices, nor to consider my behaviors. 
During this study I needed to develop this reflection as a new practice. In the relational 
counseling situation, even in brief counseling, it is usual for counselor and cUent to 
discuss the transference and countertransference. Where the focus was mathematics 
learning, would that be appropriate or necessary? In the brief counseling situation in a 
college setting, the challenge was to estabhsh for myself parameters for if, when, and 
how to use the transference and countertransference material in counseling with the 
student. I found that the following practices were appropriate in the mathematics 
counseling setting and allowed for effective use of the data from both transference and 
countertransference: 



356 

Incorporating data from instruments and from observation to consider students ' 
conscious and unconscious expectations about their relationship with the present 
mathematics teacher and tutor. With every participant, I listened for, observed, and asked 
how she experienced the present class. I asked students questions about their past and the 
present to help them discover the ways they might be appropriately or inappropriately 
bringing their past into the present.'" If it became clear that a student's experience of the 
class was discrepant from the present reality I drew her attention to it and invited her to 
consider how she might adjust to this new awareness. 

Developing reflection and self-awareness regarding countertransferential 
reactions to the tutees. I filled out the mathematics counseling session summary sheet 
after each session to help me reflect immediately on the session. I listened to my tape 
recordings of counseling sessions and study groups and studied transcripts in order to 
observe myself in relation to participants. Supervision was central in some cases to 
recognizing my countertransferential reactions. Not surprisingly, my session notes were 
often ahead of my counseling practice. 

Using self-revelation of countertransference. I found that when I did self-reveal in 
the counseling situation, both the participant and I became clearer about the relational 
patterns that might be keeping us both stuck. We were then more able to change our 
behaviors and to extricate ourselves from counterproductive patterns."" I found that I 
needed to present my experience of countertransference in a manner compatible with 
student's learning style or risk her not understanding and optimally benefitmg."'" 

Indirect use of transference and countertransference observations in situations 
where the student rejects or avoids a counseling approach (Category II: Type B students) 
or has a detached avoidant relating pattern (a Category III: Type A student in this study). 



357 

I could only talk about transference and countertransference indirectly with some 
students by noticing behaviors and perceptions and asking them to verify whether they 
were seeing the present relationships as different from the past, inviting them to notice 
the present relational reality, suggesting they evaluate the appropriateness of their beliefs 
and practices in relation to the present reality, and affirming their helpful choices and 
changes to appropriately deal with present reality. 

Supervision by a person knowledgeable in counseling. I needed a knowing 
dispassionate ear to share my actions and judgments, particularly my subjectively 
experienced transference and countertransference. Preparmg my cases for supervision 
forced me to reflect on each participant in a more global way than I had till then. 
Supervision itself provided me with affirming and challenging feedback on my progress 
thus far with each participant. It forced me to pay close attention to my own reactions and 
my tendency to impose my agenda on participants rather than facilitating their own 
choices and movement.'^^ It furnished me with possible new approaches for stuck 
situations (cf. paradoxical intention for Mulder). An even earlier supervision meeting 
may have helped me decide to do things differently from unwittingly acting out my 
countertransference.'" 
Counseling as Good-enough Tutor-Parenting 

Winnicott's concept of good-enough freed me in a number of ways to be more 
available to my tutees and to help them be strategic in their choices. I am not neutral with 
respect to procedural (only) versus conceptual (including procedural) mathematics 
pedagogy, for example. In my experience, conceptual learning helps make students 
secure in their mathematics base. The tutoring role differs from the teaching role in that 
control over the curriculum lies with the teacher not with the tutor; the tutor must support 



358 

the student in mastering the curriculum whether the tutor "approves" of the curriculum or 

not. In the context of the 10- or 15-week mathematics course where students had the 

opportunity to struggle on problems in class with coaching support from the instructor, 

the conceptual aspects were only linked with the mathematical procedures when 

individuals asked the instructor during problem-working sessions. Opportunities in 

counseling to help tutees attain a more conceptual understanding of the material were 

limited by time and content, especially with students who were akeady deeply embedded 

in a procedural approach. I found that to support a student in doing well on a 

PS YC/STAT 1 04 exam, there were times when procedural advice superceded conceptual. 

I was able to see my mathematics tutoring as good-enough ui providing for my students 

although it was less than (my) perfection. In line with good-enough parenting I also had 

to me learn to better tolerate students' mathematical goals when they differed from mine 

in contrast to my former approach of trying to badger or cajole them to take on my goals 

for them. On the other hand, I had to be carefiil not to allow this good-enough concept to 

lull me into lowermg my expectations for what they could achieve. 

Challenges and Limitations of this Approach: Integrating Counselor and 
Tutoring Roles into Mathematics Counseling 

I found that to be a good-enough mathematics counselor is very difficult. My 
"successes" from my long enculturation and experience in traditional mathematics 
teaching had only relatively recently been called into question by the nagging failures 
that drove me into my doctoral program. A cognitive constructivist, conceptual, problem- 
solving approach to teaching and learning mathematics was the solution, I was 
convinced, but I found it difficult to be that teacher, to facilitate that learning. I had 
always been the one who worked out what the problem was and structured the solution 



359 

and told the student, who ran with it, or puzzled over it, or denied it, or ignored it. I grew 
to believe that the essence of constructivism was in the student seeing the problem and, 
with the teacher as guide, finding a solution for herself, but how to be a guide? I now 
know that telling spoiled it by making it mine and not hers (even if I was "right"). Now I 
have discovered that counseling is the same. I had learned through lay counseling 
ministry training and experience that a constructivist approach was essential for healing 
and growth. That was confirmed in my doctoral psychological counseling coursework. 
Now in this study I had to integrate my emerging but tentative constructivist teaching role 
with a constructivist counseling role to be a good-enough mathematics coimselor. 
The Challenge of Learning to be a Relational Counselor 

Mulder taught me about coimseling perhaps more than any other participant 
because he would not accept my telling; he resisted it and stood up to me and I learned to 
step aside and let him fight his own battle. Not that my input was not helpful — indeed it 
was! On his own, it is almost certain that Mulder would not have made the changes he 
did but in the end they were his own changes. If I had not stepped aside he may not have 
made the final crucial change. With other participants my propensity for prescribing my 
solutions for them was not as clear to me although my experience of transference and 
countertransference gave me clues. Dr. P. saw it and helped me to begin to see it in 
supervision. With some participants, though, it was only as I analyzed the transcripts, my 
session notes, other data, my own initial analysis of the student's needs, and Dr. P.'s 
persistent supervision-style queries of that analysis that I finally heard myself telling and 
scolding and prescribing. And I finally realized how I could have done it differently, in a 
constructivist manner, because along with the telling I did some of that (work in a 
constructivist manner) and Dr. P. also pointed that out to me. All along I had the insights 



360 

and approaches of relational counseling to use; when I did use them students did well and 
found their own feet. In the end what I did with each participant (except for Kelly, Brad, 
and perhaps Autumn) was good-enough for them to gain insight into their restrictive 
mathematics relational patterns. This equipped them to make the changes necessary to 
succeed in PSYC/STAT 104. 
The Challenge of Learning to be a Relational Tutor 

I learned relational mathematics tutoring from the participants, especially Karen, 
Mulder, and Lee. I found that when I used <:onstructivist, relational counseling 
approaches such as mirroring sound thinking (even in the midst of errors or low grades) 
to build up tentative and vulnerable mathematics selves, participants began to move into a 
competence they did not know they had and then to develop that competence. When 
mstead I was drawn into participants' focus on the negatives (the errors or the low grade) 
and tried to fix it by telling the answer and teaching them more, I cut them off from that 
tentative mathematics self so that it could not grow. 

Likewise when I heard their mathematical questions and responded to their 
pressing felt needs by telling, things did not go well; when I responded by eliciting from 
them what they already knew and we went from there (e.g., parallel problem-solving), 
they grew. In the end what I did with each participant (except for Kelly and Brad) was 
good-enough for them to gain access to their growing mathematics competence, develop 
insight into counterproductive mathematical beliefs and practices, and make the changes 
necessary to succeed in PSYC/STAT 104. 

Relational counseling is based on the idea that the counselor and client are both 
adults, and the client chooses her path while the counselor supports her. The tutor-tutee 
relationship is usually an expert-novice relationship with regard to the mathematics 



361 

content and (theoretically at least) a novice-expert one with regard to the student's own 
affective experience of the mathematics. I had been learning how to negotiate the 
mathematics content in a constructivist, reciprocal way, but not the student's affective 
experience of the mathematics. In my prior tutoring practice the tutor-tutee relationship 
with regard to the student's own experience of the mathematics was more often a parent 
to child one. From this study, I found that that is the challenge for me in the practice of 
mathematics relational counseling — ^to learn how to be constructivist, non-directive, and 
supportive, while also learning from the student. This was needed not only when we dealt 
with the mathematics content, but also when we explored and gained insight into the 
affective areas of her mathematics cognition and her underlying relational patterns. 

Just as a crucial assumption of this approach is the reality of student choice and 
responsibility for choices, this assumption applies equally to the counselor. The benefit 
for the student is in helping her become conscious of her choices and the extent of her 
power to choose differently. The danger in this approach is to appear to hold a person 
responsible for things she has little power to change. Thus Jamie could choose to sign up 
for individual mathematics counseling, but the shyness and prior negative experiences 
that dominated her interactional patterns, led to her choosing to hide and disappear rather 
than relate and approach. Because she was not consciously aware that she was making 
that choice, she seemed to remain powerless to choose differently. My choice to approach 
her was perhaps going against one of the maxims of counseling (Wait for the person to 
seek your help; that will mean she is ready and willing to receive it.) but because my 
choice was good-enough in this case, Jamie became aware of her choices and her power 
to chose differently. At other times my choices to be parent rather than peer with the 



362 

student were not good-enough (cf. Brad and Autumn). My awareness of my own power 
to choose my roles and the importance of my choices grew as the study proceeded. 

BRIEF RELATIONAL MATHEMATICS COUNSELING: A SUMMARY 

The traditional model for providing mathematics academic support typically 
excludes from consideration many aspects of the student's relationship to mathematical 
learning and compartmentalizes what is considered into content knowledge, and some 
aspects of affect. Tobias' concept of mathematics mental health provides a different 
perspective for viewing the struggling mathematics student. A brief relational counseling 
approach prioritizes students' mathematic mental health problems and provides a means 
of dealing with them in a hoUstic and productive way, without ignoring or minimizing 
important elements. My categorization scheme uses both the relational and cognitive 
diagnosis as a way of understanding and dealing with complexity. 

It is important to highlight here crucial ways that this new approach differed from 
more traditional approaches to mathematics support. But first I must point out that this 
study alone, while it puts forward considerable evidence of student change, does not 
provide quantifiable comparisons between the effects of this approach and the effects of 
traditional mathematics support. For comparisons, the sample was small, there was no 
matched sample to receive control treatment (traditional mathematics support) and 
changes reported and the processes that led to these changes were in many ways not 
quantifiable nor easily verifiable. Nevertheless the differences I observed were striking. 

Needy students do not necessarily access traditional mathematics academic 
support. Jamie would likely not have opted for academic support unless meeting with a 
tutor were a course requirement or there was a class-link tutor she could get to know first. 
Jamie's dilemma presented me a disturbing possibility that I had previously only vaguely 



363 

considered — there are likely unknown numbers of students who might benefit from this 
course intervention but would never come to a traditional Learning Assistance Center; 

Class-linking provides unique observation opportunities for more thorough and 
effective counseling. Being able to observe student behaviors and practices in the 
classroom'"' provides the tutor with data for timely and focused counseling interventions. 
If Jamie had come to a traditional Learning Assistance Center, the tutor would not have 
had the benefit of observing her in class, nor have been aware of how important that 
observation was. In my former role I would likely have worked with Jamie on her 
mathematics and found it to be relatively sound. We would not have foimd the real root 
of her mathematics learning issues nor explored ways she could see herself differently as 
a successful mathematics learner. Other students in the study who needed assistance also 
indicated that they would not have accessed my help if I, as the tutor, had not been in the 
classroom.'"" 

In contrast with current learning support experience, early and thorough 
diagnostic assessment of both mathematics cognition and mathematics relational issues 
is possible. Such timely diagnosis is key to growth in mathematics skill and improved 
mathematics mental health for students who are willing to explore both. Even for students 
who do not wish to explore their mathematics relational problems, their assessments can 
be used to design mathematics-only counseling interventions that assist them to make 
academic and indirectly, relational progress; 

Without a relational counseling approach focused on the student 's transference 
and the counselor 's countertransference, the counselor might not be able to identify and 
deal with students ' core mathematics mental health issues. If I had not identified Jtunie's 
transference towards Ann and me as frightening, dangerous teachers, I might have acted 



364 

out the same kind of countertransference that Ann did (staying away in order not fulfill 
Jamie's fears). Instead I broke through and came close and was not dangerous; 

For counseling to be efficacious, the importance of the classroom emotional 
climate established by the mathematics instructor cannot he underestimated. The positive 
emotional climate established by the instructor in this study created an environment for 
most students where damaged attachments to mathematics teachers could be repaired, 
where underdeveloped or vulnerable mathematics selves could grow, and where no 
further damage was done. By contrast, counseling students taking a course where they 
experience the instructor and the classroom as indifferent or abusive would likely have to 
take a different direction and would invariably have reduced efficacy in achievement and 
emotional healing for the student. 

A counseling use of the mathematics addresses the various mathematics mental 
health problems caused by the ways the mathematics content has been and is being 
taught. Because a procedural approach to mathematics is closely related to conceptual 
linking difficulties (i.e., linking procedures with their conceptual base) and a tendency 
towards learned helplessness in the mathematics course environment, individualized 
mathematics-focused counselmg approaches (e.g., mathematics course management and 
conceptual problem-solving counseling) may be called for. One use of exam analysis 
counseling is to help negative students break a negative focus by affirming or mirroring 
sound mathematical thinking, thus building up their underdeveloped selves. Conceptual 
linking counseling offers students a secure mathematics base they may not have 
previously experienced. Such intentional uses of mathematics tutoring as counseling 
contrast with traditional uses and demonstrate promise for improving the student's 
mathematics mental health. 



365 

LIMITATIONS OF STUDY 

It is important to evaluate the conclusions of the study in terms of the sample, the 
measurement instruments and their uses, and the research methodology. 

Sampling limitations. The number of participants in the study was small and they 
were taken from a small urban New England commuter university. The nature of the 
study necessitated a small sample but the fact that students were from widely varying 
backgrounds enhances its value. The small size of the sample restricted the use of 
quantitative results of the instruments to descriptive support for qualitative results within 
the sample, aiding the ongoing counseling process, and understanding of individual 
outcomes. The findings from the instruments may not be generalizable to students 
enrolled in other mathematics or statistics courses nor attending other types of college or 
university although uses of some instruments (e.g., the Metaphor and the JMK Affect 
Scales) seem applicable for counseling purposes in any setting. 

Quantitative instrument limitations and uses. The only quantitative instrument 
reliably calibrated on large samples was the Algebra Test (see Appendix C). Apart from 
the first 20 questions of the Feelings Survey that were taken from the 98 item MARS 
which was normed thirty years ago (Suiim, 1972), all instruments with quantitative 
outcomes, except for the class exams and other class evaluations, were created or adapted 
and the results evaluated by the researcher. Individual results were compared with those 
of the (small) class group and individual changes are described in comparison with other 
researcher-observed changes and class achievement changes. 

Researcher bias. The participants' words and actions were filtered through 
researcher bias. Relational data were collected via counseling session interaction where 
the researcher and the participant were working for change, and the interactions did 



366 

change both. Although ongoing analysis of the interactions by the researcher (with 
participant feedback) and clinical supervision (and later evaluation of researcher analysis 
by the clinical supervisor) were designed to monitor, interpret, and neutralize this bias, 
interpretations of the data by others might yield different conclusions about the relational 
outcomes of the study. 

Possible omission of important student factors. My understandings of key aspects 
of mathematics functioning (cognition and affect)'"'", were applied, integrated, and 
adapted in this study. I adapted Mitchell's relational conflict theory and his concept of 
three dimensions a person's relationality, and revised Tobias' five tier categorization of 
college mathematics students. In addition to Dweck, Seligman and Beck's work on 
learned helplessness and depression, the researcher's own findings were also used to 
ground the study. None of these, separately or together, has been used in a holistic study 
of college students' mathematics mental health or of interventions to improve it while the 
student was taking a college mathematic course. Because of this it is certainly possible 
that important aspects of students' mathematics mental health were not addressed. All 
findings in this study should be interpreted in this light. 

RECOMMENDATIONS FOR FURTHER RESEARCH 

Based on fmdings in this study and the limitations, I make the following 
recommendations for future research: 

Counselor characteristics subtype B students and counselor-student match. My 
use of brief relational mathematics counseling helped students from all three categories 
identified. Students who benefited less were from subtype B of both Category II (Autumn 
and Mitch) and Category III (Kelly) — students whose ways of dealing with the 
vulnerability and under-confidence of their mathematics selves were relatively inflexible 



367 

or unstable. Counselor characteristics may have been a factor in this relative lack of 
success. Additional research needs to be done when offering this counseling to such 
students. It will be vital to investigate counselor characteristics, counselor-student match 
and interventions that may help them succeed. 

Quasi-experimental studies. My adaptation of Mitchell's (1988) relational conflict 
theory to mathematics support in this study yielded an understanding of the three 
dimensions of participants' relationality and their central relational conflicts that enabled 
us to resolve that conflict well enough in the brief time available for them to be 
successful in the course (with the exceptions noted above). It is not possible to say what 
their outcomes would have been if they had not participated in the counseling 
intervention. Additional research using this approach with other college mathematics 
students, comparing their process and outcomes with those of matched samples of 
students who receive tutoring support only, and with matched samples of students who 
receive no support would further our understanding and test its generalizability, 
particularly the finding of increased achievement and improved mathematics mental 
health. 

Gender differences. I found interesting gender differences in the way similar core 
problem were expressed. Men and women, especially those within Category II: Type A 
and Category III: Type A groups, whose core problems were the same, differed markedly 
in their emotional conditions, practices and ways of relating. With Category II: Type A 
students, the women expressed their insecure attachment to mathematics with anxiety; the 
man expressed his with a frantic and counterproductive attempt to learn it all. With 
Category III: Type A students, the woman expressed her underdeveloped mathematics 
self with underconfidence, empty depression and hostile detachment; the men expressed 



368 

theirs with overconfidence, unrealistic bravado (grandiosity) and resistance. Further 
research on gender differences within and across types may be called for to confirm (or 
disprove) that the presentation of similar core problems consistently differs predictably 
according to gender. 

Counselor-student match and gender. I had considerable difficulty in overcoming 
my countertransference reactions to grandiose men of Category III: Type A. I wanted to 
deflate their inflated sense of prowess. My reactions seemed to come at least in part from 
our gender difference. One of these men, who spoke of his difficulties working with 
women, made only minimal progress in counseling. I wonder if a male counselor may 
have been more successful in supporting and developing his underdeveloped mathematics 
self. Research into the effects of counselor-student match by gender could shed light on 
this. 

Identifying mathematics situational depression. This study suggests that some 
students may suffer from mathematics situational depression. The College Learning 
Metaphor (pre-and post) and repeated use (administered at every session) of the JMK 
Mathematics Affect Scales analyzed together seemed to aid diagnosis and help to monitor 
this condition. Further, it can alert the counselor to a need for a specific and timely 
intervention. Ongoing research using both the College Learning Metaphor Survey and the 
JMK Mathematics Affect Scales conducted with large numbers of students would 
investigate a possible relationship. Such a finding would investigate the simultaneous use 
of both instruments to rule out mathematics depression and aid accurate diagnosis 

Mathematics relational counseling and other classroom conditions. This study 
was conducted in the context of a classroom where the instructor created a positive 
relational climate, where the mathematical demands were somewhat more procedural 



369 

than conceptual, and where the conceptual content was taught by lecture discussion and 
the mathematics procedures developed in problem-working sessions. Relational 
counseling in other contexts is likely to look different and have different outcomes for 
different categories of student. Classes may differ in how mathematics is taught: they 
may stress non-routine problem-solving; they may be designed to challenge student 
misconceptions; they may involve mathematics procedures only being demonstrated by 
the teacher on the board. Classes may differ in relational climate: the teacher may be 
disdainful of what she perceives to be students' low ability and poor understanding; she 
may be judgmental of certain student approaches; she may ignore or insuk student who 
struggle. Further studies of the use of relational counseling to support students in 
different settings accordmg to how mathematics is taught and according to relational 
climate would contribute to our understanding of the efficacy and limits of its use. 

CONCLUSION 

I learned during the summer of 2000 to open my learning specialist eyes wider 
and to see through the lens of relational conflict theory. This at once complicated and 
clarified my task. New complexities arose in having to look now not only at the student's 
mathematics tasks but also at her whole approach to the mathematics course, her 
mathematics self, her intemaUzed presences, and her patterns of mathematics 
interpersonal interactions. 

In this study I determined it was indeed possible for a mathematics learning 
specialist with some exposure to the field of psychological counseling to hoUstically help 
traditional and non-traditional aged college students taking an introductory level college 
mathematics course. Most came to understand their mathematics learning issues and 
found their own coherent explanatory frame for how the aspects of mathematics 



370 

cognition were personally configured within their relational history. Students became 
conscious of their areas of embeddedness as well as how they contributed to their own 
immobility, they made changes, and they improved their mathematics mental health. 
They attained "good-enough" success in the current course and some even seemed to 
develop the heart to tackle future mathematical challenges more effectively. 

I found that the concept of relationality with its three dimensions did provide an 
adequate frame for me to understand and focus on each participant's particular relational 
conflicts and I found that the approaches of relational and cognitive therapy were useful 
as elements of an approach designed to address those difficulties. The approach 
highlighted my role in the counselor-student dyad and I found that to the extent that I 
reflected on how I reacted and interacted with the student, the insights I gained led me to 
change in ways that promoted student growth. In sum, I found that the relational conflict 
perspective has given me a new, more nuanced, and authentic way of seeing students and 
helping them to see themselves and their interactions, in the academic support and course 
classroom setting. 



371 



' As this is an even numbered ciiapter, I use "shie," "her," and "hers" for third person generic pronouns. 

" For example, I agreed with the Karen who said she probably could not do it and at first ignored the Karen 
who expressed and showed surprising competence; I agreed with the Mulder who said he could do it and 
dismissed the Mulder who expressed frustration and struggle. 

'" Each of these students (Autumn, Jamie, Lee, and Mitch) was ambivalent about her own mathematical 
ability. None saw him/herself in the category of "some people can do math" (question 9, Part 11, 
Mathematics Beliefs survey) although all but Jamie initially believed her "ability in mathematics" could 
improve (question &, Part II). The Algebra Test is not a test of ability; it shows a student approximately 
where she is on a developmental path. I was wary of feeding into any fixed trait beliefs about mathematical 
ability even the "I'm one of those people who can do mathematics" belief that saw others as not being able 
to. In feet, I took every opportunity to dispel such fixed trait beliefs. I deemed a developmental view that 
saw growth of self and others as always possible and expected improvement in relation to intelligent effort 
as much healthier. In the context of a 10 week college mathematics course, however, being at a higher 
algebra concept developmental level certainly gave a student an advantage over a student at a lower level. 

" Two of these five did not exhibit an adequate operaticm sense (Pierre's was inadequate and Lee's was 
marginal), one did not exhibit adequate open ended arithmetical thinking /problem-solving (Autumn was 
inadequate), and one did not exhibit an adequate large (>I000) integer number sense (Jamie' was 
marginal). 

" I found the AFS assessment to be too long, with a number of questions not relevant to this statistics course 
or discerning enough. The graph related questions not varied or discerning enough, there were not enough 
operation sense questions, and relevant categories such as order of operations were not addressed. In 
addition, within each category, the questions were not designed developmentally to reveal levels of 
understanding. 

" For example, Jamie's "fear of the storm coming back" metaphor was not principally about the 
mathematics itself but about a dangerous classroom environment with dangerous teachers. It took us some 
time to link this with Jamie 's own behaviors in the storm — staying inside in order to keep safe in this 
dangerous situafion. Mulder himself linked his "Fox Mulder searching for aliens" metaphor with making 
mathematics hard for himself, so that became our initial focus — the ways Mulder did mathematics that 
made it hard for him. We at first missed the link for Mulder between the object of the search — aliens — and 
mathematics. An important piece to Mulder's difficulties was that he was indeed seeing mathematics as 
alien, so he was using alien search techniques to master it ratha- that exploring and mastering it logically 
and conceptually. 

"" Mulder's metaphor was Fox Muldw seardiing for aliens and Brad's was Sherlock Hoknes trying to crack 
a mystery. 

"" Each had an average score of 3.5 or above on a scale of I through 5. 

^ This social anxiety was related to and complicated by Jamie's fear of too much success that invited 
attention, expectation, and pressure from ha- father for fiiture performance. 

" Jamie answered 4 on her pretest on this but on her posttest she answered: 3 (not in front of class, 
individual work), 4 (our size class) and 5 (math lecture size class like at State University) 

" I did this somewhat unevenly. For example, I indulged Lee because we were pals (staying wdth her 
transference and my countertransference); I did not become conscious enough of how her positive feelings 
towards me contrasted with her negative feelings toward the instructor so I did not help her to evaluate that 
against the reality and we did not ask ourselves how our reactions might have a detrimental effect on how 
she approached PSYC/STAT 104. We should have, for example, questioned the discrepancy between her 
effort in the class and on homework versus how much time she was spending with me. 



372 



^ For example, with Mulder, I told him I felt like his scolding mother — that helped me in my process of 
moving out of that inappropriate role; I could ha\e done it more clearly with Karen, e.g., "When I listen to 
what you say about yourself doing mathematics I feel depressed, but when I look at the mathematics you 
are doing and the ways you are changing I admire you and am hopefiil that you can learn it and get a good 
grade in the class." I believe that this would have had a positive effect on the development of her 
underdeveloped mathematics self 

™ For example, with Robin a visual learner with auditory processing difficulties, when I told her of my 
dual reactions: that in class she seemed to be acting as an intellectually incompetent female and I felt 
irritated and alarmed at the same time, whereas one-on-one in the mathematics counseling situation I found 
her to be intellectually competent and 1 admired and respected her ability to manage the content, she did not 
understand because I did not use visual pictures that she had introduced in her metaphor of the ditzy village 
girls versus Belle. 

™ During supervision my strong fiiistration with Autumn became apparent and my judgment of her self- 
containment, her emotional distance, "1 want to shake her" (Jillian, July 20, 2000). I found myself reporting 
a very similar reaction to Brad, but more for his refijsal (from my perspective) to face the reality of his 
situation, and his bravado in the face of the realities. With both of these participants 1 had been very 
directive, partly the cause, Dr P gently suggested, for their digging in and my fi-ustration. His reminding me 
that it was their motivations for change or stasis that needed to be revealed and respected and not my 
motivations for them imposed. He encouraged me to find what in me had been triggered by their behaviors 
and attitudes. 

'" In my session notes, I found that my insights and determinations were at times ahead of my actions. 
Earlier supervision might have alerted me to those discrepancies and led to a different approach with Brad 
or a timely intervention with Kelly. 

What being a class-link contributes to the efficacy of the brief relational mathematics counseling model 
is: Counselor presence in the central context, leading to 

1. Participation by the students in the counseling, 

2. Inside perceptions of the instructor and the Course that could be used in counseling, and 

3. In-class perceptions of student practices that could also be used in counseling 

""^ Seven of eight participants who finished the course and who responded to my post-study survey 
(December 2000) acknowledged that they would not have accessed mathematics learning assistance at the 
Learning Assistance Center if I had not been in the class with them. Of these seven, three, Karen, Mitch, 
and Pierre, said they will now access the Learning Support Center in the future when they are taking a 
mathematics course, one of these though, Karen, only "...as long as Jillian is thae because you really 
helped me in my last math class." Another, Lee, indicated a very conditional, "if I do [take another math 
course], and I find it difficult, 1 will use the Learning Assistance Center facilities if 1 think they will best 
help me." The others. Autumn, Jamie, and Mulder, said they probably would not access the Learning 
Support Center in the future when they are taking a mathematics course, Jamie because she is "kind of shy" 
and a "helper" who doesn't "really like to ask for help," and Mulder because "Unless I am really struggling 
I do not go for help. I like to figure things out on my own." Autumn gave no explanation. 

'^" Suggested by Schoenfeld (1992), Dweck (1975; 1982; 1986), Tobias (1993), Skemp (1987), Boaler 
(1997) and others. 



373 



374 



Appendix A: Table Al 


Proposed Brief Relational Mathe matics Counseling 






Relational 


Student's Mathematics History 


Mathematics Affect Now 


Mathematics Cognition Now 


Relational Diagnosis 


Proposed Treatments 


Dimension 
















How does this develop? 


What can go wrong? 


Expected Affective symptoms 


Expected Cognitive symptoms 


Expected Central Mathematics Relational 
Conflicts 




Mathematics 


1 . Mirroring 


1 . Neglect . . . chronic lack 


I . empty depression; learned 


1 . sound mathematics preparation 


Self: Conflict between conscious 


For Self. Counselor 


Self 


2. Idealization of teacher 


of mirroring => 


helplessness 


2. adequate mathematics 


ambition/desire to succeed in course and 


mirrors studait's 




image 


underdevelopment of self: 


2. grandiosity 


preparation 


underlying belief in inability to succeed in 


mathematics self. 




3. (a) Internalization and 


low self-esteem, under- 




3. underpreparation 


course (low/undermined self-esteem) 


provides self for 




transformation of teacher 


con fidence 








idealization. 




image into values and 


2. Failure to provide 








provides manageable 




ideals... superego, (b) 


realistic curbing of 








frustrations to push 




manageable frustration => 


grandiosity => 








student to 




development of student's 


underdevelopment of self: 








development & 




competence. 


low self-esteem. 








realization of 




confidence and basic 


ambivalent/over- 








competence 




mathematics self-esteem: 


confidence 












healthy narcissism 












Mathematics 


1. Installation of bad 


1 .Experience of 


1. guilt, shame => depression 


1 . intern al ized presences 


Internalized Presences: Conflict between 


For Bad Internalized 


Internalized 


internalized teacher 


endangerment by bad- 


2. fear of judgment => anxiety 


supportive or at least not 


conscious desire to and perhaps belief in 


Presences: 


Presences 


presence in the unconscious 


enough teacher => moral 




detrimental to mathematics self 


self for success in course and 


Counselor provides 




(Note: degree of badness) 


conversion to self as bad 




and internalized mathematics 


Internalized presences insisting that one is 


self(and points to 




2. Identification of the ego 


internalized presence or 




values (superego) 


bad/cannot succeed 


instructor) as good 




with the bad internalized 


repression of bad 




2. internalized presences 




replacements for bad 




presence (Note: extent of 


internalized teacher object 




undermining mathematics self 




presences and 




identification) => 


2. Experience of 




3. punishing mathematics 




refutes claims of bad 




development of defenses to 


mathematics as punitive 




superego ("internal saboteur") 




internalized 




protect the ego from these 


internal saboteur: 




making mathematics self feel 




presences 




bad internalized presences 


superego => sense of 
moral failure 




guilt/ shame 






Mathematics 


Mathematics teachers: 


Mathematics teachers: 


1 . grief/loss => 


1 . sound mathematics attachment 


Attachments: Conflict between conscious 


For Compromised 


Attachments 


1 . Teacher provides good- 


Teacher unavailable 


depression 


2. traumatized mathematics 


desire to succeed in course and detached 


Attachments: 




enough caregiving: 


and/or unresponsive => 


2. separation anxiety from teacher 


attachment 


attachment pattern that prevent studait 


Counselor provides 




responsive & available => 


student develops insecure 


and/or mathematics 


3. failure of mathematics 


from getting the help he/she needs or 


self(and points to 




teacher as secure base 


attachment to teacher: 




attachment 


dependent relational pattern that prevents 


instructor if 




2. Student develops secure 


anxious, ambivalent, or 






student from taking responsibility with 


applicable) and 




attachment => able to 


disorganized attachment 






support or ambivalent unstable 


promotes 




explore and return to secure 


Mathematics: 






attachment pattern 


mathematics as 




base when needed 


Teacher does not know 








secure teacher and 




Mathematics: 


and/or teach mathematics 








secure mathematics 




1 . Teacher has good-enough 


well enough=> student 








bases on which 




grasp of ftindamental 


develops anxious. 








student can rely 




arithmetic/ transitions to 


ambivalent, or 












algebra/algebra 


disorganized attachment 












2. Teacher promotes 


to arithmetic and/or 












mathematics rather than self 


algebra 












as authority for correctness 














3. Teacher believes in 














student's prowess and 














provides developmentally 














appropriate mathematical 














tasks; student has necessary 














tools => student develops 














secure attachmait to 














mathematics 




















1 







375 



APPENDIX B 

Individual Mathematics Counseling 
Assessment and Treatment Tools 



1 . Mathematics Counseling Session Reflection 

2. Student Mathematics History Interview Protocol 

3. College Learning Metaphor Survey 

4. Negativity/Positivity Survey 

a. JMK Mathematics Affect Scales 

b. JMK Mathematics Affect Scales, revised 

5. Survey Profile Summary Sheet 

6. One-On-One Mathematics Counseling Evaluation 



376 



1 . Mathematics Counseling 

Session # 

Name Course/Semester_ 

Counselor Date 

Notes 



Transference/Countertransference 



Self 



Object 

Space-in-between 

Summary: 



Thoughts for the next Session: 



©Jillian M. Knowles, UNHM, Summer 2000 



377 



MATHEMATICS HISTORY INTERVIEW PROTOCOL 

NAME AGE 

DATE MAJOR 



1 . Tell me how you usually feel when you are doing mathematics. 



In the class you are in now, how does it feel to be: 

• in class? 

• doing homework? 



• doing an exam? 

2. Describe your best experience doing mathematics? 



Why; 



3. Describe your worst experience doing mathematics? 



Why*^ 



4. A. Have you always felt this way about doing mathematics? [If not, when 
and why and how did how you feel doing mathematics change?] 



5. Is doing mathematics the same as or different from doing other 
subjects for you? How? 



6. Do you do mathematics outside of classes — when do you do it in your 
daily life? 



378 



7. Tell me how well you do in mathematics courses, in daily life. 



8. What is mathematics anyway? How would you describe it to a friend? 



9. If doing mathematics is different for you from doing other activities, 
why do you think that is so? 



10. How important do you think math/stats will be for you in your future? 
How does that make you feel? 



11. Are any parts of math comfortable for you to do? Tell me a little about 
it... 



12. What are your least favorite types of mathematics to do? Tell me a little 
about it... 



379 



13/14/15 In Elementary/Middle/HIgh school, what was mathematics like 
for you? 

What type of mathematics did you do? 



What tools did you use? 



Do you remember the teacher? 



Was there anything about you/your family that you felt made a 
difference in how the teacher treated you/her expectations of you? 



How did you get through school math when it got hard? [when you felt 
unable to do it well] 



Did you receive any extra help? How was that for you? 



How do you think math should have been taught/the learning 
environment should have been for you to have done better in it? 



How do you think YOU could have done things differently in 
mathematics for you to have done better in it? 



380 



16. List the mathematics courses you took in high school, the year you took 
each, and the grade you earned in each: 

Mathematics Course Year (e.g., 1995) 



1th 



9'" Grade 
10"^ Grade 
11 '''Grade 
12"^ Grade 



Grade earned 



1 7. Did your parents work with you with math at home? How was 

that? Theu- attitudes to math? to you doing math? Any brothers? Or sisters? 



© Compiled by Jillian Knowles, Summer 1997; revised Summer 2000 



381 

3. College Learning Metaphor Survey: 

The College Learning Metaphor was administered at the beginning of the first 
session to all participants. When possible, it was also administered in the final session to 
assess any changes (but see also One-on-One Counseling Evaluation below). 

COLLEGE LEARNING METAPHOR SURVEY 
Name Date 

1. Make a list of metaphors that show how you FEEL about 

MATHEMATICSA^OURSELF DOING MATHEMATICS. For 
example, if it were a color what color would it be? If it were weather or 
an animal or a fictional character or . . . what would it be? 



Now choose one of the metaphors from 1. that most closely describes 
your relationship with MATHEMATICS and write more about why this 
metaphor describes your relationship with MATHEMATICS. 



3. As you reread your metaphors, what do they tell you about your attitudes 
as you do MATHEMATICS? your expectations of yourself doing 
MATHEMATICS? your predictions about your success in 
MATHEMATICS? 



Adapted from; Gibson, H. (1994). "Math is like a used car": Metaphors reveal attitudes towards 
mathematics. In D. Buerk (Ed.), Empowering students by promoting active learning in 
mathematics (pp. 7-12). Reston, West Virginia: National Council of Teachers of Mathematics. 

© Jillian Knowles, revised December 1999. 



382 



4 a. 

JMK Mathematics Affect Scales 
Name Date 



On this questionnaire is a group of scales. Please read each scale carefully. Then 
indicate the part of each scale which best describes the way you have been feeling while 
doing mathematics during the PAST WEEK . INCLUDING TODAY . If an interval on 
the scale better describes your range of feelings rather than point, indicate that range with 
a line. If the words on the scale do not accurately describe your feelings, supply your 
own. 

1 . When I think about doing mathematics, 

I tend to put work ofif: 

never a lot 



sometimes 



2. If I think about how I experience my problems with mathematics, 

I tend to feel discouraged: 

never very much 



sometimes 



3. When I think about my mathematics future, 

I feel: I feel: 

confident hopeless/nothing can 

improve 



4. When I think about the mathematics course I am taking now, 

I: I: 

would withdraw if I 
like it could 



5. When I think about how I do mathematics, 



383 



I: 

feel pride in 
how I do it 



I: 

feel ashamed/ _ 
all the time 



6. When I think of my mathematical achievements, 

I: 

feel satisfied 



I feel 
discouraged 



I: 

feel like a complete 
failure/ 



7. While I am doing mathematics, 

I can: 

make 

mathematical 

decisions on my 

own 



I get 
confused 



I can: 
not make 
mathematical 
decisions on my own 



©Jillian Knowles, Lesley College, MA, 2000. Not to be used without permission. 



384 



4. b. I revised the JMK Mathematics Affect Scales following the study by adding an 
eighth scale to gauge responder's sense of himself in relation to the rest of the class. See 
chapter 6 for discussion. 



8. When I compare myself with others in my mathematics class, 

lam: 

better at 
mathematics 
than most of them. 



I am about the 
same level as most of them 

©Jillian Knowles, UNHM, Fall 2001 Not to be used without permission. 



I am: 

much worse 
at mathematics 
than most of them. 



385 



5. As a way of integrating students data and using it for ongoing insight and intervention 
in counseling, I placed an individual's scores with the class extreme scores for each scale 
and discussed the concepts and implication with participants during counseling sessions. 
See chapter 5 and 6 for discussion. 



Name 



Not at all 

U 



1 

Not at all 

U 



Survey Profile Summary Sheet 

Class Pre/Post 

MATHEMATICS FEELINGS 
Math Testing Anxiety 



Number Anxiety 



Date 



very much 

— ► 



very much 

— ► 



1 



lU 



Not at all 
U 



1 



Abstraction Anxiety 



very much 

— ► 



III 



MATHEMATICS BELIEFS SURVEY 
Procedural Math Conceptual Math 



4 ► 



Toxic /Negative 



Learned Helpless 

* 



1 
OVERALL SUMMARY 
Negative 



Healthy/Positive 



4 > 



Mastery Orientated 

► 



5 
Positive 



■« ►• 



©Jillian Knowles, Lesley College, Summer 2000 



386 

5. One-On-One Mathematics Counseling Evaluation. Participants were asked to respond 

to whether and in what ways they had changed in their approach to mathematics learning 

during the course and to write about factors to which they attributed any changes. As part 

of this they were invited to write a different metaphor if a new one was applicable. See 

chapters 6 and 8 for a discussion of responses. 

I administered this to individual counseling participants during class posttesting, July 31. 

2000 _______^ ^^____ 

One-On-One Mathematics Counseling Evaluation 
Name (optional) Date 



Please answer the following questions as honestly as possible from your point if view. 

Please be open with any criticisms, questions or suggestions you have. Use the back if 

necessary. 

1 (a) What was your initial motivation for signing up to meet with me for one-on-one 

mathematics counseling? 



(b) Did that motivation change? If so, how and why? 



2. Did the way you see yourself as a mathematics learner change in any way as you were 
doing PS YC/STAT 1 04this summer? If so, in what ways did you change? Did 

your math metaphor change? To what? What, do you think, were the main factors in 
that change? (e.g., the way the class was taught?, the professor? the testing style?, 
meeting with me?, the math content? a personal change? ...a combination?) 



3. Do you think your meetings with me affected how you were approaching 
PS YC/STAT 104? 

If so, in what ways? 



4. Do you think your meetings with me affected your success in PSYC/STAT 1 04? If 
so, in 

what ways? 



387 



5. With regard to Question 4, how do you.define "success in PSYC/STAT 1 04" ? 



6. How. if at all, do you think your overall experience in PSYC/STAT 104this summer 
will affect how you will approach your next mathematics-related course or 
challenge? 



©Jillian Knowles, Lesley College, Cambridge, MA, Summer 2000 



388 



Appendix C 
Class Assessment and Observation Tools 

1 . Beliefs Surveys: 

a. Modified Mathematics Beliefs Survey, June 5, 2000 

b. Modified Mathematics Beliefs Survey, Revised Version, August 2002 

c. My Orientation to Mathematics Survey, Short Revised Version, May 
2003 

2. Mathematics History, Feelings and Evaluations Surveys: 

a. Pretest Mathematics Background Survey and My Mathematics Feelings 
survey 

b. Pretest Mathematics Background Survey Revised Version, August 2002 

c. Posttest Course Reflection and Evaluation Survey that preceded the 
posttest My Mathematics Feelings survey 

d. Class-Link Evaluation. 

3. Arithmetic Assessment: 

a. Arithmetic for Statistics (AFS) Assessment 

b. Arithmetic for Statistics (AFS) Assessment Profile form 

4. Statistical Reasoning Assessment (SRA) 

5. The Algebra Test and sample scoring sheet 

6. Observation Tools: 

a. Music Staff Class Interaction Observation Chart 

b. Class Layout Observation Form 

c. Problem Working Session Interaction Chart (Class 13) 

d. Class Summary analysis sheet 



389 

1 . Belief Surveys: 

When Ema Yackel, with counseling psychologist Ann Knudsen (and later 
Carolyn Carter) developed and ran a mathematics anxiety reduction course, they aimed at 
challenging and changing students' procedural, helpless, and mythical beliefs about 
mathematics and themselves and reducing anxiety levels while the students learned 
conceptual mathematics usmg a problem-solving, constructivist approach (E. Yackel, 
personal communication, January 2 1 , 2000; Carter & Yackel, 1 989). 

Yackel created a two-part mathematics beliefs survey as a before and after 
instrument for the course. The fu-st part assesses beliefs about mathematics along a 
continuum from beliefs about mathematics as conceptual (Skemp's (1987) "relational" 
mathematics) through mathematics as procedural (Skemp's "instrumental" mathematics). 
In the second part Yackel had included questions that she felt from her experience as a 
mathematics educator to be important for a healthy approach to mathematics, questions 
she "found interesting" (personal communication, January 21, 2000). Because Carter and 
Yackel used Kogelman and Warren's (1978) anxiety reduction approach in their 
workshops, I reviewed Kogebnan and Warren's Ust of myths and used in my survey ones 
related to the topics I surveyed: 

1 Men are better at math than women. 

2 Math requires logic, not intuition. 

3 You must always know how you got the answer. 

4 Math is not creative. 

5 There is always a best way to do a math problem. 

6 It's always important to get the answer exactly right. 



390 



7 It's bad to count on your fingers. 

8 Mathematicians do problems quickly, in their heads. 

9 Math requires a good memory. 

10 Math is done by working intensely until the problem is solved. 

1 1 Some people have a math mind and some don't. 

12 There is a magic key to doing math. 

These beliefs can be grouped into three broad categories: Some of these myths 
relate to an erroneous or procedural view of mathematics and self (e.g., 2, 3, 5, 6, 
8, and 9); some relate to learning style bias and constricted /jerfagogy (e.g., 3, 4, 
7); others are embedded in American cultural tradition (1, 11, 12). Yackel's 
survey used versions of myths 1, 4, 5, 6, 9, 11 isolated by Kogehnan and Warren. 
I added question 19 (Part II) that Kogelman and Warren isolated (cf their #7) and 
Yackel had not included. 

I also added some perceived usefulness questions (Part II questions 22 
and 23) to touch on Sherman and Fennema's (and others') usefulness factor found 
to be related to mathematics learning motivation and achievement although 
Yackel had already included two usefukiess items. Yackel's survey touched on 
male domain and mathematics-related affect factors identified on Fennema- 
Sherman Attitude scales and I added a parent/teacher item (Part II #21) that I 
believed may be linked to learned helplessness. Yackel and I did not include any 
success items (Fermema & Sherman, 1976; Mulhem & Rae, 1998). 



391 

In order to elucidate student's beliefs around their control of the situation 1 modified 
the second part, adding questions related to learned helplessness (Licht &Dweck, 1984) 
such as: 

7. 1 think my ability to do mathematics can improve. SD D U A SA 

(SD means "strongly disagree," SA means "strongly agree,"), that asked whether the 
respondent has a fixed trait mathematics theory about herself or not. 

A number of questions relating to learned helplessness were already in the first part 
since, for example, a belief in mathematics as procedural sees the mathematics as outside 
one's control, leading to a helpless response if one does not "recognize" the problem or if 
one "forgets" the procedure. I thus created a Learned Helplessness through Mastery 
Orientation (LM) Scale within the larger scales. There were fourteen questions that 
pertained to student beUefs and behaviors on this continuum. (Since these questions are 
embedded in the larger survey I labeled the Learned Helpless/Mastery Oriented questions 
as LM and signed them LM- to indicate a Learned Helpless and LM + to indicate 
Mastery Orientated belief or behavior respectively. The label LM -and the LM+ signs did 
not appear in the student administered version of the Modified Mathematics Belief 
Scale.) 

During post analysis as I looked for factors that discriminated among the categories 
of students I identified (see chapter 7), I found that questions m this Beliefs Survey ( Part 
I, Items 4, 7, 9, and 10) that related to achievement motivation contributed to that 
identification. Since these questions are embedded m the larger survey I labeled the 
performance/learning achievement motivation questions as P/L and signed them P/L - to 
indicate a performance achievement motivation and P/L + to indicate learning 



392 

respectively. The labels P/L -and the P/L + did not appear in the student administered 
version of the Modified Mathematics Belief Scale. 

Thus my Modified Mathematics Beliefs Systems Survey yielded three measures of 
belief and attitude: 

1 . mathematics as procedural through conceptual, 

2. mathematics learning approaches and attitudes as toxic through healthy, and 

3. learned helpless through mastery orientation, and 

to provide a starting point for discussion, challenge, and reeducation m the mathematics 
counseling setting, and a fourth: performance through learning achievement motivation, 
to aid post analysis . 

I used this information with each participant by discussing their positions on their 
individualized Surveys Profile Summary (see Appendix B), by investigating individual 
item responses, and by explaining the concepts involved and their ramifications to 
mathematics learning. 

I gave the Mathematics Beliefs Systems Survey as a posttest to ascertain if any 
changes had been made over the summer. I had opportunity to discuss these changes with 
only one participant, Jamie. See chapters 3, 6, and 7 for further discussion. 



393 



1. a. Modified Mathematics Beliefs Systems Survey administered as a pre- and posttest to 
the class on June 5. 2000 and on July 3 1 , 2000 respectively. 



Modified Mathematical Belief Systems Survey 

Date 



Name/Number 

All individual responses to this survey will be kept strictly confidential. Your responses 
will be used to study relationships among student beliefs about mathematics, past 
teaching methods used, effects of mathematics learning assistance and certain other 
variables such as mathematics background. 
For each item, circle the response that indicates how you feel about the item as indicated 

below. PLEASE add your own comments or questions at any point in the Survey. 



Strongly 








Disagree 


Disagree 


Undecided 


Agree 


SD 


D 


U 


A 



Parti 

1. Doing mathematics consists mainly of using rules. 

2. Learning mathematics mainly involves memorizing procedures SD D U 
and formulas. 

3. Mathematics involves relating many different ideas. 

4. Getting the right answer is the most important part of P/L- 
mathematics. 



Strongly 








/\gree 
SA 








SD D 


U 


A 


SA 


SD D 


u 


A 


SA 


SD D 


u 


A 


SA 


SD D 


u 


A 


SA 



5. In mathematics it is impossible to do a problem unless 
you've first been taught how to do one like it. 



LM- 



6. One reason mathematics is so much work is that you need to 
learn a different method for each new class of problem. LM- 

7. Getting good grades in mathematics is more of a motivation 
than is the satisfaction of learning the mathematics content. P/L- 

8. When I learn something new in mathematics I often continue 
exploring and developing it on my own. LM+ 

9. 1 usually try to understand the reasoning behind all the rules 
I use in mathematics. P/L+ LM+ 



SD D U A SA 



SD D U A SA 



SD D U A SA 



SD D U A SA 



SD D U A SA 



lO.Beingable to successfully use a rule or formula in mathematics SD D 
is more important to me than understanding why and how it 
works. P/L- 



U A SA 



11 . A common difficulty with taking quizzes and exams in 

mathematics is that if you forget relevant formulas and rules 
you are lost. LM- 



SD D U A SA 



SD 


D 


U 


A 


SA 


SD 


D 


u 


A 


SA 


SD 


D 


u 


A 


SA 


SD 


D 


u 


A 


SA 



394 



12. It is difficult to talk about mathematical ideas because all you SD D U A SA 
can really do is explain how to do specific problems. 

13. Solving mathematics problems frequently involves SD D U A SA 
exploration. 

14. Most mathematics problems are best solved by deciding SD D U A SA 
on the type of problem and then using a previously learned 

solution method for that type. LM- 

15. 1 forget most of the mathematics Ilearn in a course soon SD D U A SA 

after the course is over. LM- 

1 6. Mathematics consists of many unrelated subjects. 

1 7. Mathematics is a rigid uncreative subject. 

18. In mathematics there is always a rule to follow. 

19. 1 get frustrated if I don't understand what I am studying 
in mathematics. 

20. The most important part of mathematics is computation. SD D U A SA 



Part II 

1 . I usually enjoy mathematics. 

2. Mathematics is boring. 

3. When I work on a difficult mathematics problem and I can't 

see how to do it in the first few minutes, I assume I won't be 
able to do it and I give up. LM- 

4. When I read newspaper and magazine articles I skip over SD D U A SA 
numbers, graphs, and numerical material. 

5. I only take mathematics courses because they are required.LM- SD D U 

6. I think mathematics is fiin and is a challenge to learn. LM+ SD D U 

7. I think my ability to do mathematics can improve. LM+ SD D U 

8. Mathematics/statistics, in my experience, has no connection SD D U 
to the real world. 

9. Mathematics is a subject that some people can do and others SD D U A SA 
can't. LM- 

10. My overall feeling towards math is positive. 

11. Mathematics is used on a daily basis in many jobs. 

1 2. Mathematics is easy for me. 



SD 


D U 


A 


SA 


SD 


D U 


A 


SA 


SD 


D U 


A 


SA 



A 


SA 


A 


SA 


A 


SA 


A 


SA 



SD 


D U 


A 


SA 


SD 


D U 


A 


SA 


SD 


D U 


A 


SA 



SD 


D 


U 


A 


SA 


SD 


D 


u 


A 


SA 


SD 


D 


u 


A 


SA 


SD 


D 


u 


A 


SA 


SD 


D 


u 


A 


SA 


SD 


D 


u 


A 


SA 



395 



13.1 like to work on hard mathematics problems. LM+ 

14. Most mathematics courses go too fast for me. 

1 5. Mathematics is a subject men do better in than women. 

1 6. 1 would like to learn more about mathematics/statistics. 

17.1 was better at Geometry than at Algebra. 

18. 1 have to understand something visually before I can "get' it 
auditorily/verbally 

19. 1 think having to use fingers or other calculating SD D U A SA 

manipulatives is childish and shows you are not very good 
at mathematics. 

20. 1 have avoided/delayed taking a mathematics class because of SD D U A SA 
my worry about my ability to succeed in it. LM- 

21. 1 have had a math teacher/guidance counselor/parent who SD D U A SA 

has made me feel I did/do not have the ability to take higher 
level math classes. 

22. I'll need mathematics/statistics in my future schooling. 

23. I'll need mathematics/statistics in my future work. 

Other Comments and Questions: 



©Adapted, with permission, by Jillian Knowles, Lesley College, Cambridge MA, Summer 2000, for the 
purposes of her Doctoral Dissertation Research, from Ema Yackel's 1984 Survey created for a Purdue 
University Continuing Education Reducing Mathematics Anxiety course 



SD D U 


A SA 


SD D U 


A SA 



396 



1 . b. Modified Mathematics Beliefs Survey, Revised Version, August 2002, recommended 

for use after analysis of dissertation data. 

Modified Mathematical Belief Systems Survey 
Name/Number Date 









Strongly 


isagree 


Undecided 


Agree 


Agree 


D 


U 


A 


SA 



All individual responses to this survey will be kept strictly confidential. Your responses 
will be used to study relationships among student beliefs about mathematics, past 
teaching methods used, effects of mathematics learning assistance and certain other 
variables such as mathematics background. 
For each item, circle the response that indicates how you feel about the item as indicated 

below. PLEASE add your own comments or questions at any point in the Survey. 

Strongly 

Disagree 
SD 
Parti 

1 . Doing mathematics consists mainly of using rules. SD D U A SA 

2. Learning mathematics mainly involves memorizing procedures SD D U A SA 
and formulas. 

3. Mathematics involves relating many different ideas. SD D U A SA 

4. Getting the right answer is the most important part of SD D U A SA 
mathematics. P/L- 

5. In mathematics it is impossible to do a problem unless SD D U A SA 
you've first been taught how to do one like it. LM- 

6. One reason mathematics is so much work is that you need to SD D U A SA 
learn a ditferent method for each new class of problem. LM- 

7. Getting good grades in mathematics is more of a motivation SD D U A SA 
than is the satisfaction of learning the mathematics content. P/L- 

8. When I learn something new in mathematics I often continue SD D U A SA 
exploring and developing it on my own. LM+ 

9. 1 usually try to understand the reasoning behind all the rules SD D U A SA 
I use in mathematics. P/L+ LM+ 

lO.Being able to successfiilly use a rule or formula in mathematics SD D U A SA 
is more important to me than understanding why and how it 
works. P/L- 

1 1 . A common difficulty with taking quizzes and exams in SD D U A SA 
mathematics is that if you forget relevant formulas and rules 

you are lost. LM- 

12. It is difficult to talk about mathematical ideas because all you SD D U A SA 



SD 


D 


U 


A 


SA 


SD 


D 


u 


A 


SA 


SD 


D 


u 


A 


SA 


SD 


D 


u 


A 


SA 



397 



can really do is explain how to do specific problems. 

13. Solving mathematics problems frequently involves SD D U A SA 
exploration. 

1 4. Most mathematics problems are best solved by deciding SD D U A S A 
on the type of problem and then using a previously learned 

solution method for that type. LM- 

15. 1 forget most of the mathematics I learn in a course soon SD D U A SA 

after the course is over. LM- 

16. Mathematics consists of many unrelated subjects. 

17. Mathematics is a rigid uncreative subject. 

18. In mathematics there is always a rule to follow. 

19. 1 g e t fru s trated if I don't und e rstand what I am studying 

in math e matics . Item broken into two parts and moved to Part II 

20. The most important part of mathematics is computation. SD D U A SA 



Part II 

1 . I usually enjoy mathematics. SD D U A S A 

2. Mathematics is boring. SD D U A SA 

3. When I work on a difficult mathematics problem and I can't SD D U A SA 

see how to do it in the first few minutes, I assume I won't be 
able to do it and I give up. LM- 

4. When I read newspaper and magazine articles I skip over SD D U A SA 
numbers, graphs, and numerical material. 

5. I only take mathematics courses because they are required.LM- SD D U A SA 

6. I thinlc math e matics is fun and is a chall e nge to l e arn. LM+ SD D U A SA 

7. 1 think my ability to do mathematics can improve. LM+ SD D U A SA 

8. Mathematics/statistics, in my experience, has no connection SD D U A SA 
to the real world. 

9. Mathematics is a subject that some people can-de understand SD D U A SA 
and others can't. LM- 

1 0. My overall feeling towards math is positive. SD D U A SA 
(new) I rate my ability in mathematics as: 

poor; below average; average; above average; excellent (circle one) 

1 1 . Math e matics is used on a daily basis in many jobs . SD D U A SA 



398 



12. Mathematics is easy for me. SD D U A SA 

(new) Pt 1, 19 (a) I am able to learn mathematical procedures SD D U A SA 

(no score on scale) 
(new) Pt I, 19 (b) I do not expect to be able to understand what SD D U A SA 

I am doing in mathematics or why 
13. 1 like-teworii en-hard on mathematics problems until LM+ SD D U A SA 
I master them. 

14. Most mathematics coxirses go too fast for me. 

15. Mathematics is a subject men do better in than women. 

16. 1 would like to learn more about mathematics/statistics. 

17. 1 was b e tt e r at Geom e try than at Alg e bra. 

18. I have to understand s om e thing visually b e for e I can "get' it 
auditorily/verbally 

19. 1 think having to use fmgers or other calculating SD D U A SA 

manipulatives is childish and shows you are not very good 
at mathematics. 

20. 1 have avoided/delayed taking a mathematics class because of SD D U A SA 
my worry about my ability to succeed in it. LM- 

21.1 have had a math teacher/guidance counselor/parent who SD D U A SA 

has made me feel I did/do not have the ability to take higher 
level math classes. 

22. I'll need mathematics/statistics in my future schooling. 

23. I'll n ee d math e matic s /statistics in my fiitur e work. 

Other Comments and Questions: 



SD 


D 


U 


A 


SA 


SD 


D 


u 


A 


SA 


SD 


D 


u 


A 


SA 


SD 


D 


u 


A 


SA 


SD 


D 


u 


A 


SA 



SD 


D U 


A SA 


SD 


D U 


A SA 



©Adapted, with permission, by Jiliian Knowles, Lesley College, Cambridge MA, Summer 2000, for the 
purposes of her Doctoral Dissertation Research, from Ema YackePs 1984 Survey created for a Purdue 
University Continuing Education Reducing Mathematics Anxiety course 



399 



I.e. My Orientation to Mathematics Survey, Short Revised Version, May 2003 

This is a shorter Learned Helpless-Mastery Oriented focused Revised Version of 
Modified Beliefs Survey. This shortened form includes items from the Beliefs Survey that 
investigate students' beliefs about (a) the nature of mathematics (conceptual versus 
procedural), (b) ability and effort beliefs and attributions items (that include some U.S. 
cultural beliefs), (c) student mathematics practices items, and (d) achievement motivation 
items, but no usefulness beliefs, or mathematics attractiveness attitudes items. I have 
added a category of items (e) to investigate student social practices related to social 
learned helplessness in accessing support (see chapter 6 and 7). 

MY ORIENTATION TO MATHEMATICS LEARNING 
Name Course Date 



SD = Strongly Disagree; D = Disagree; N = Neutral; Agree; SA = Strongly Agree 

1 . In mathematics it is impossible to do a problem unless SD D N A SA 
you've first been taught how to do one like it. 

2. One reason mathematics is so much work is that you need to SD D N A SA 
learn a different method for each new class of problem. 

3. When Ilearn something new in mathematics I often continue SD D N A SA 
exploring and developing it on my owti. 

4. 1 usually try to understand the reasoning behind all the rules SD D N A SA 
I use in mathematics. 

5. A common difficulty with taking quizzes and exams in SD D N A SA 

mathematics is that if you forget relevant formulas and rules 
you are lost. 

6. Most mathematics problems are best solved by deciding SD D N A SA 
on the type of problem and then using a previously learned 

solution method for that type. 

7. 1 forget most of the mathematics I learn in a course soon SD D N A SA 

after the course is over. 

8. When I work on a difficult mathematics problem and I can't SD D N A SA 

see how to do it in the first few minutes, I assume I won't be 
able to do it and I give up. 

9. I only take mathematics courses because they are required. 

1 0. I think my ability to do mathematics can improve. 

1 1 . I rate my ability to do mathematics as: (circle one)poor| below |average|above lexcellent 

{average | |average| 

12. Mathematics is a subject that some people can do and Others SD D N A SA 
can't. 



SD D N 


A SA 


SD D N 


A SA 



SD 


D 


N 


A 


SA 


SD 


D 


N 


A 


SA 



400 



13. I can do mathematical procedures. 

14. 1 don't expect to be able to understand what 1 am doing in 

mathematics or why. 
15. In the past, working hard has not changed how I did in SD D N A SA 

mathematics. 

16. 1 work on hard mathematics problems until 1 master them. SD D N A SA 

17. If I get a good grade in mathematics it is only because I work SD D N A SA 
hard, not because I am smart. 

18. I delay taking mathematics classes because of my worry SD D N A SA 
about my ability to succeed in them. 

19. If I got a bad grade in mathematics it is only because I didn't SD D N A SA 

work hard. 

20. Teachers should not pick out particular students to answer SD D N A SA 
questions in class. 

21. I would never volunteer to answer a question a teacher asked SD D N A SA 

in class even if I knew the answer. 

22. If I didn't understand what the professor was saying about a math problem I would 

a) ask her in class SD D N A SA 

b) go to her office hours to ask her SD D 

c) ask a student sitting near me SD D 

d) go to the Learning Center to ask a tutor SD D 

e) do nothing and hope it would be covered in the next class SD D 

23. Getting the right answer is the most important part of SD D 
mathematics. 

24. Getting good grades in mathematics is more of a motivation SD D N A SA 
than is the satisfaction of learning the mathematics content. 

25. Being able to successfully use a rule or formula in mathematics SD D U A SA 
is more important to me than understanding why and how it 

works. 



N 


A 


SA 


N 


A 


SA 


N 


A 


SA 


N 


A 


SA 


N 


A 


SA 



1. 


SD 


D 


N 


A SA 




5 


4 


3 


2 1 


2. 


SD 


D 


N 


A SA 




5 


4 


3 


2 1 


3. 


SD 


D 


N 


A SA 




1 


2 


3 


4 5 


4. 


SD 


D 


N 


A SA 




1 


2 


3 


4 5 


5. 


SD 


D 


N 


A SA 




5 


4 


3 


2 I 


6. 


SD 


D 


N 


A SA 




5 


4 


3 


2 1 


7. 


SD 


D 


N 


A SA 




5 


4 


3 


2 1 


8. 


SD 


D 


N 


A SA 




5 


4 


3 


2 I 


9. 


SD 


D 


N 


A SA 




5 


4 


3 


2 1 


10. 


SD 


D 


N 


A SA 




1 


2 


3 


4 5 


11. 


P 


b/a 


av 


a/av e 


12. 


SD 


D 


N 


A SA 




5 


4 


3 


2 1 


13. 


SD 


D 


N 


A SA 


14. 


SD 


D 


N 


A SA 




5 


4 


3 


2 1 


15. 


SD 


D 


N 


A SA 




5 


4 


3 


2 1 


16. 


SD 


D 


N 


A SA 




1 


2 


3 


4 5 


17. 


SD 


D 


N 


A SA 




5 


4 


3 


2 1 


18. 


SD 


D 


N 


A SA 




5 


4 


3 


2 1 


19. 


SD 


D 


N 


A SA 




1 


2 


3 


4 5 


20. 


SD 


D 


N 


A SA 


21. 


SD 


D 


N 


A SA 


22 e) 


SD 


D 


N 


A SA 




5 


4 


3 


2 1 


23. 


SD 


D 


N 


A SA 




5 


4 


3 


2 1 


24. 


SD 


D 


N 


A SA 




5 


4 


3 


2 1 


25. 


SD 


D 


N 


A SA 




5 


4 


3 


2 1 



401 



Score My Mathematics Orientation 
©Jillian Knowles, Lesley University. 
Fall 2002. 
Not to be used without permission 



TOTAL: 



Learned Helpless 



21 



Mastery Orientated 



42 



63 



84 



105 



402 



Subscales From MY ORIENTATION TO MATHEMATICS LEARNING 
Scale 1 : Mathematics as: 



Procec ural 



Total from Questions I, 2, 5, 6, 7: 



Cone ;ptual 
1> 



10 



15 



20 



25 



Scale 2: My mathematics practices as 
Learned Helpless 



t 



Total from Questions 3, 4, 8, 9, 16, 21: 
Score 21. SD D N A SA 
5 4 3 2 1 



P lastery Orientated 



12 



18 



24 



30 



Scale 3: My beliefs about my mathematics self as 
Detrirnental 



13. 



SD D N A SA 

12 3 4 5 



6 12 

Total from Questions 10, 1 1, 12, 13, 14, 18: 

Score II. p b/a av a/av e 

2 2 4 4 3 



Constructive 



18 



24 



30 



Scale 4: Attributions as: 

Unhealthy 



Total from Questions 15, 17, 19: 



Hea thy 



12 



15 



Scale 5: Social/ Accessing Support as: 
Independent 



14 
Total from Questions 20, 2 1, 22 a, 22b, 22c, 22d, 22e: 



Self-reliant 



21 



2S 



Score 



20. 



21. 



22a. 



22b. 



22c. 



22d. 



22e. 



SD 


D 


N 


A 


SA 


5 


4 


3 


2 


1 


SD 


D 


N 


A 


SA 


5 


4 


3 


2 


1 


SD 


D 


N 


A 


SA 


1 


2 


3 


4 


5 


SD 


D 


N 


A 


SA 


1 


2 


3 


4 


5 


SD 


D 


N 


A 


SA 


1 


2 


3 


4 


5 


SD 


D 


N 


A 


SA 


1 


2 


3 


4 


5 


SD 


D 


N 


A 


SA 


5 


4 


3 


2 


1 



Scale 5: Achievement Motivation as: 
Performance 



Learning 

♦■ 



16 



2b 



Total from Questions 4, 23, 24, 25: 



403 

2. Mathematics Feelings Surveys. 

The Mathematics Anxiety Rating Scale (MARS) has been normed and is perhaps 
the most used in the field (Richardson & Suinn, 1 972). It is long however (98 items), 
only yields one measure, but seems to address anxiety in a number of different settings 
that it would be helpful to differentiate. Rounds and Hendel did a factor analysis of 94 of 
the items of MARS and identified 30 items that they found measured two relatively 
homogeneous factors (15 items each) they called "mathematics testing anxiety" and 
"numerical anxiety respectively" (Rounds & Hendel, 1 980). Ron Ferguson created a 
three-factor instrument from this using the twenty items that loaded most heavily on these 
factors (10 each) and adding ten items to measure a factor he labeled "abstraction 
anxiety" to make an instrument more applicable to a college setting. Factor analysis 
showed that his items did measure a factor different from the two that Rounds and 
Hendel identified (Ferguson, 1986). I have slightly changed some of Ferguson's items 
and adopted his instrument, calling it Measuring Mathematics Feelings rather than 
Ferguson's suggestive "Phobus" (a moon of Mars and the root of the word phobia). 
Fergsuson has placed his items in the public domain and I have purchased MARS (adult 
form) from Dr. Suinn so that I could use the 10 mathematics testing anxiety and the 10 
numerical anxiety MARS items that Ferguson used from Rounds and Hendel' s factor 
analysis. Ferguson's instrument is not normed but, as my primary use of it is in the 
counseling situation, its ability to quickly assess three pertinent factors of a student's 
anxiety, two of which relate to the type of mathematics, thus providing a point of 
discussion, made it more usefiil for this study than the full MARS. The principle reason 
for assessment in this study was not to compare an individual or group with equivalent 



404 

people in the wider population, but to compare an individual with herself as she made 
changes. 

I used this information with each counseling participant by discussing their 
positions on their individualized Survey Profile Summary (see Appendix B), by 
discussing individual item responses, and by explaining in more detail the concepts 
involved. 

I gave the Measuring Mathematics Feelings as a post test to ascertain if any 
changes had been made over the summer. I had opportunity to discuss these changes with 
only one participant, Jamie. Discussing her changes on the instrument highlighted 
another aspect of anxiety that was particularly pertinent to her — ^the interaction of social 
anxiety with the mathematics learning or performance situation. It was this element of 
Jamie's mathematics anxiety that had been reduced over the summer. 

I have therefore coded each item of Measuring Mathematics Feelings as: 

1 . Position in relation to others: 

a. P for primarily public, 

b. S for solitary, 

c. S/P for solitary with a public component, 

d. P/S for public with a solitary component, depending on the relational 
setting implied or explicitly referred to in the question, and 

2. Setting of activity 

a. CI to indicate primarily classroom setting for the activity and 

b. Cl/H for an activity that occurs both at home and in the classroom. 



405 



I have done this to aid analysis with the student responder and for post analysis. 
My coding may change in discussion with a student who feels the question situation as 
more or less public or more or less solitary. For iurther discussion see Jamie and Me 
chapter 6. 

Note: Dr. Richard Suinn has given me permission to include here as samples (to 
be used by readers only with his permission) ten of the twenty items that I took from his 
Mathematics Anxiety Rating Scale (MARS) that form parts I and II of the My 
Mathematics Feelings survey. I have deleted the other ten items but retained the above 
categorization of them. 



2. a. Measuring Mathematics Feelings Pretest Survey, administered in class, June 5, 
2000. This survey also includes the Pretest Mathematics Background Survey questions. 

Statistics in Psychology PSYC 402, Summer 2000 
Please fill in whatever of the foUowmg you feel comfortable sharing. All the data will be 
kept confidential. Participation or non-participation in this study will not affect your 
grade in this class in any way. 



406 



Name/Number Date 

Major Is this class required for your major? If yes, why do 

you think it is required? 

Last math class taken before this one Year Grade 



What statistics have you studied before?. 

What, in your opinion is the relationship between mathematics and statistics?_ 



What grade do you hope for in this class? What grade do you expect?_ 

Measuring Mathematics Feelings 



Each question below describes a mathematics-related activity or situation. Please 
indicate on the scale of 1 through 5 how much you are scared by that mathematics-related 
activity or situation nowadays. 

N 
I. S. Signing up for a math course.* 

2.P/ClWalking into a math class.* 

3. P/CLRaising your hand in a math class to ask a question.* 1 

4. S/P/Cl.Taking an examination (fmal) in a math class.* 

5.S.** 

6.S.** 

7.S/P/C1 Waiting to have a math test returned.* 

8.S.** 

9.S. Receiving your final math grade in the mail.* 

lO.S/P/Cl** 

* Sample items from the Mathematics Anxiety Rating Scale. The Mathematics Anxiety Rating Scale 

(MARS) is copyrighted by Richard M. Suinn, Ph.D. Any use of the MARS items requires the permission of 

Dr. Suinn: suinn(Silam ar.colostate.edu . I retained these items because class and/or individual response 

changes on them over the course were notable (see chapter 8 for fiirther discussion) 

** The items from MARS used are omitted here as per agreement with Dr. Richard Suinn. 
PART II 

Not at all Very much 
1 .P. Determining the amount of change you should get back from a purchase involving 
several items.* 12 3 4 5 



tall 

1 2 


3 


4 


Very much 

5 


1 2 


3 


4 


5 


1 2 


3 


4 


5 


1 2 


3 


4 


5 


1 2 


3 


4 


5 


1 2 


3 


4 


5 


1 2 


3 


4 


5 


1 2 


3 


4 


5 


1 2 


3 


4 


5 


1 2 


3 


4 


5 



407 



2.P. Listening to a salesperson show you how you would save money by buying his 
higher priced product because it reduces long-term expenses.* 

12 3 4 5 

3.P.** 12 3 4 5 

4.S. Reading your W-2 form (or other statement showing your annual earnings and 
taxes).* 

12 3 4 5 

5.P. 12 3 4 5 

6.P. Hearing iriends make bets on a game as they quote the odds.* 

12 3 4 5 

7. P/S.** 12 3 4 5 

8. S. ** 12 3 4 5 

9. S. ** 12 3 4 5 
lO.S. ** 12 3 4 5 



* Sample items from the Mathematics Anxiety Rating Scale. The Mathematics Anxiety Rating 
Scale (MARS) is copyrighted by Richard M. Suinn, Ph.D. Any use of the MARS items requires 
the permission of Dr. Suiiin: suinn(g),lamar.colostate.edu . I chose to retain these items because 
they elicited the highest anxiety responses or changed most over the course. 
** The items from MARS used are omitted here as per agreement with Dr. Richard Suiim. 



408 

PART III 

Not at all Very much 

l.S/Cl. Having to work a math problem that has x's and v's instead of 2s and 3s. 

12 3 4 5 

2.P/C1. Being told that everyone is familiar with the Pythagorean Theorem. 

12 3 4 5 

3.S/P/C1. Realizing that my psychology professor has just written some algebraic 
formulas on the chalkboard. 12 3 4 5 

4. S/Cl. Being asked to solve the equation x^ - 5x + 6 = 

12 3 4 5 

5. P/Cl. Being asked to discuss the proof of a theorem about triangles. 

12 3 4 5 

6. S/Cl. Trying to read a sentence full of symbols such as: SSxx = Sx^ - (£x)^ 

N 

12 3 4 5 

7. P. Listening to a friend explain something she just learned in calculus. 

12 3 4 5 

8. S, CVH. Opening up a math book and not seeing any numbers, only letters, on an 
entire page. 

12 3 4 5 

9.S. Reading a description from a college catalog of the topics to be covered in a math 
course. 

12 3 4 5 

lO.P, Cl/H. Having someone lend me a calculator to work a problem and not knowing 
which button to push to get the answer. 12 3 4 5 

The 98 item Mathematics Anxiety Rating Scale (MARS) was developed by Richardson and Suinn in 1972 
(Richardson & Suinn, 1972). Ron Ferguson created Phobus(a moon of Mars) by first choosing 20 items 
from MARS, ten found by Rounds and Hendel to be related to Mathematics Test Anxiety (Part I) and the 
other ten to be related to Number Anxiety (Part 11) (Rounds & Hendel, 1980). Ferguson then added ten 
more items to measure what he calls Abstraction Anxiety (Part 111) (Ferguson, 1986, 1998). 1 have slightly 
changed items 2 and 3 of Part II and 6 and 7 of Part III. 
©JiUian Knowles, Lesley College, Summer 2000. 



409 



2. b. Following the study I found that there were a number of background details I had 
failed to ascertain from participants. To remedy this I sent an e-mail survey to which 
most participants responded. The following is my revision of the Pretest Mathematics 
Background Survey that I would recommend to avoid these difficulties I encountered. 
Revision additions are bolded. Revision deletions are shown as strike throughs. 

Statistics in Psychology PSYC 402, Summer 2000 
Please fill in whatever of the following you feel comfortable sharing. All the data will be 
kept confidential. Participation or non-participation in this study will not affect your 
grade in this class in any way. 
Name/Number Date 

Major Is this class required for your major? If yes, why do 

you think it is required? 

Last high school math class taken Year (e.g., 1997) Grade (e.g.. A) 

Last college math class taken Year(e.g., 1997) Grade (e.g.. A) 

Have you ever repeated a mathematics course? If so what course and when? 

Are you repeating PSYC/STAT 104? 



Have you taken the Brookwood State University mathematics placement test? 
If yes, what mathematics course was recommended? 



Did you take that course? If yes, when and what grade (e.g., B) 

What statistics have you studied before?. 



What, in your opinion i s th e r e lationship b e tw ee n mathematics and statistics? 



What grade do you hope for in this class? What grade do you expect?_ 

Describe your worst experience in a mathematics class? 



How old were you? 



Describe your best experience in a mathematics class 



How old were you? 



410 



2 . c. The Posttest Course Reflection and Evaluation survey questions preceded 

Measuring Mathematics Feelings Posttest Survey, administered in class, July 31, 2000. 

The Measuring Mathematics Feelings part was identical with that on the Pretest Survey 

so it is not included here. _____^ 

Statistics in Psychology PSYC 402, Summer 2000 
Please fill in whatever of the following you feel comfortable sharing. All the data will be 
kept confidential. Participation or non-participation in this study will not affect your 
grade in this class in any way. 
Name/Number Date 

Please describe how taking this summer course, PSYC 402, was for you? 



What did you learn about yourself as a mathematics learner doing this course?_ 



How much time did you spend on studying/homework per week on the course? 



What about the statistics covered in this class is still puzzling to you?_ 



Will you try to fmd out more about it? How?_ 



What is the most meaningfiil concept/idea you learned about statistics in this class? 



Why?_ 



What grade did you hope for in this class? What grade are you getting? 

How satisfied are you with this grade? With what you learned? With your own approach? 



411 

2. d. Class-Link Evaluation. This is an evaluation form designed by Learning Assistance 
Center personnel for students to evaluate the class-link tutor and the instructor's use of 
her. 1 administered this form to the class during posttesting, July 3 1 , 2000. See chapters 8 
for discussion. 



futor's Name: 



Qass Link Evaluation 

Course: 



Instructor's Name:_ 



Yean 



[ ]FaU 



[ ] Spring 



1. How often did you see a tutor for this course? 



2. Describe how the tutor worked with you. For example, did s/he demonstrate? Ask questions? Read aloud? 



. 3. In what specific ways was the tutor helpful? 



4. Were there areas where you feel s/he could have been more helpful? If so, how? 



5. Did the tutor ever confuse you? If so, how? 



Please answer the following questions by circling the most descriptive response. 
The scale ranges from strongly disagree (1) to strongly sgiiee (5). 



Qass links are an asset to a class. 
12 3 4 5 

I would prefer to deal with the instructor rather than the 

class link. 

12 3 4 5 

I sometimes feel the instructor used the class link in order 
to avoid student contact. 

12 3 4 5 

I do not like having another student involved in my work. 
12 3 4 5 

The class link was knowledgeable about the course 

content. 

12 3 4 5 

I was usually able to get in contact with my class link when 

a need arose. 

12 3 4 5 

The class link was reliable in keeping appointments. 

12 3 4 5 

The class link should have been better informed about the 
requirements and materials of the course. 

12 3 4 5 



\ 



The class link was easy to talk with. 
12 3 4 5 

My work iinproved through my association with the class 
link. 

12 3 4 5 

The instructor and the class link communicated 

sufficiently. 

12 3 4 5 

The class link made me feel comfortable in the learning 

process. 

12 3 4 5 

I sometimes felt that the class link was too critical. 
12 3 4 5 

It is generally helpful to have a class link with whom to 

discuss ideas. 

12 3 4 5 

I feel that the instructor relied too much on the class link. 
12 3 4 5 

ThxrkyQufcfrtcJimgthetmBto<xmfktethkeidi0ticn Please 
add cany other aomTBnts you inish to make aba-it dx tutaifs) and/or 
the Learning Center on the lack (fthisjbrm 



413 

3. I developed the Arithmetic for Statistics instrument during the course as a diagnostic 
for participants in response to the type of arithmetical reasoning errors I saw and lack of 
the type of arithmetical reasoning that if used might have led students to correct their 
errors. I used operation sense and number sense questions suggested by Marolda and 
Davidson (Marolda & Davidson, 1994), items used by Liping Ma in her assessment of 
elementary teachers' profound understanding of fundamental arithmetic (Ma, 1999a), a 
proportional reasoning question, number line scale questions investigating small 
(decimal) and large numbers, some normal curve area under the curve and horizontal 
scale questions, a coordinate graph question and a pie graph question. Some questions 
were open-ended; others closed; some asked for a written explanation. 1 was able to use it 
with some of the participants during the course but others completed it during the post- 
testing session after the MINITAB project presentations in the second to last class 
meeting (July 3 1 , 2000) and others mailed theirs to me. Robin and Brad did not complete 
theirs. For fiuther discussion see chapters 6, 7, and 8. 



414 



This diagnostic was administered during individual counseling sessions with some 
students and given to all the students who had not already taken it during the July 3 1 , 

2000 post- test session after the MINITAB project presentations. 

Arithmetic for Statistics 
Assessment 
Name Date 



1. When you multiply 61.2 and 3.5 the product is 21.4; 264.2; 2,142 or 214.2? 

2. When you divide 12 by 0.12 you get a number smaller than 12? A number smaller than 
1? A number larger than 12? 

3. When you multiply, you always get a number bigger than the one you started with? 
Yes/No Explain. 

4. When you divide, you always get a number smaller than the one you started with? 
Yes/No Explain. 

5. If you earn 10% interest per year on your investment of 

$ 1 million, how much would you earn? 
$1 billion, how much would you earn? 
$1 thousand, how much would you earn? 
$1 hundred, how much would you earn? 
$10.00, how much would you earn? 
Now work out your earnings if the interest rate is 8% 

6. Does Va lie between 7/12 and 2/3? Explain? 

Explain. 



8. Given that 1 is the largest probability you can get, what could you say about a 
probability of 0.099? 0.99? 0.119? 



9. Is .099 closer to 1 or to 0? Explain. 

10. a. Which is a better sale, 2/5 off or 40% off or .04 off? Why? 



415 



b. In a group of 48 students, 1 out of 8 is of African origin, 2 out of 8 is Latino, and 4 
out of 8 is of European origin, and the rest are of Asian origin. How many students are 
there from each racial category in the whole group? 



1 1 . On this line place the point 9.9. 



01 2 3456 789 10 



12. On this line, place the points 9 and 0.9 and 0.09 and 0.009 



01 2 3456 789 10 



13. On this line place the point 0.99 



01 2 3456 789 10 



14. On this line place the point 4.19 



01 2 3456 789 10 



15. On this line place the point 3.99 



01 2 3456 789 10 



416 



16. On this line place the point 6.49 



1 



7 8 9 10 



17. What fraction of the area under the curve is colored yellow? What percent? What 
amount, given that the ,^ — ' [ ~\ total area under the curve is 1 unit"^ 




What fraction of the area under the curve is colored ^|? What percent? What 
amount, given that the total area under the curve is 1 unit? 

18. Z = -1.645. Where should it be on this Standard Normal Graph? 




-2-1012 



19. Fill in the missing number labels for the points on the line; 



0.02 




417 



20. Fill in the missing number labels for the points on the line 



2.15 



21. a. For the following normal distribution of continuous data, fill in the missing number 
labels for the points on the line. 



^ = 25 A 
CT = 2.5 / 




^ 


\ 
















X 




-2-1012 Z 

b. How is this standard normal distribution graph related to the one above with ^ = 25 
and a = 2.5? 



22. Fill in the missing number labels for the points on the line: 



0.1 



0.6 



23. Place the points 1 .85and -1 .85 on this number line. 



-2-10123 
24. Fill in the missing number labels for the points on the line: 



108 



108.45 



418 



25. From the function graph below find the value of Y for which the X value is 5. 
Think of a situation in which one variable is related to another in the way shown on the 
graph below. Fill in the table with data of all the points shown on the ftinctiop line. 

X 




X 



10 



26. On each of the following three number lines think of three different numbers 
appropriate for that scale and plot them. 



■1000 



-500 



500 



1000 



-10 



10 



-.1 -.05 .05 .1 

Create a scale on this number line to plot these numbers and then plot them: 25, 150 



Create a scale on this number line to plot these numbers and then plot them: 0.04, 0.45, 
3.05 



Create a scale on this number line to plot these numbers and then plot them: 1 800, 85 




27. Block out 0.35 of this pie graph. How much is left? 

28. The pie graph below represents the population of 1,500 students at a small liberal arts 
college. 35% are freshmen; 25% are sophomores; 25% are juniors and the rest are 
seniors. How many are in each class? Show them on the pie graph. 60% of each of the 
freshman and sophomore classes are women. 44% of the junior class and 40% of the 
senior class are men. Create a chart to show the make-up of the college by gender. Show 
it on the other pie or other graph. 





© Jillian Knowles, Lesley 
University, Summer 2000 



420 



3. b. 




_'s Arithmetic for Statistics Understanding Profile 




Course: 




Date: 






Numb 


Statistic 


Small 


Large 


Proper 


Place Value/ 


Operation 


Open Ended 


Overal 


er 


al sense 


(<1000) 
Integer 


Integer 
Number 


Fractional 
Sense 


Decimal/ Percent 
Sense 


Sense 


Arithmetical 
thinking/ 


1 level 


Correc 




Number 


sense 








problem 




t 




sense 










solving 






17a(3) 


5c. 


5a. 


6. 


a <5 


1. 


3b 






17b(3) 


5d. 


5b. 


7. 


8(3) 


2. 


4b 






18 


5e. 


51(2). 


10a. 


9. 


3 


6b 






19b(3) 


5f(3). 


26a(3) 


17a(3) 


10a 


4. 


7b 






21a(5) 


26b(3) 


26f(2) 


17b(3) 


12a 


7. 


8(3) 






21b 


26d(2) 






12b 




9b 






25(6) 








12c 
12d 
13 

14 
15 
18 

19a(4) 

20(4) 

22(4) 

23(2) 

26c(3) 

26e(3) 




25(1) 

26a(3) 

26b(3) 

26c(3) 

26d(2) 

26e(3) 

26f(2) 

27(2) 

28(6) 




Ta >5 

1. 

11 




























16 


















19b(3) 


















21a(5) 


















24(4) 








Totals 


22 or 16 


11 


9 


9 or 3 


la|<5:33 lal>5:15 


5 


33or 

27w/o28 






















% 


















Not 


















Attempted 


















Comments 



















©Jillian Knowles, Lesley University, Cambridge, MA, June, 2001 



421 



4. Statistical Reasoning Assessment. I was aware at the beginning of the study of a 
number of factors that that led me to conjecture that the chief aim of PSYC/STAT 104 
would not be to change students' misconceptions about probability and statistics or to 
develop their statistical reasoning. Instead, I supposed the aim would be to use a 
traditional lecture and test approach to have students become familiar with standard 
means of sorting and describing data (descriptive statistics) and with recognizing when 
and knowing how to use standard parametric and nonparametric statistical analysis to test 
hypotheses about populations (inferential statistics), that is, to introduce potential social 
scientists to procedures they would later use to do their own research (see chapter 5). 
This is not to say that these two aims are necessarily incompatible but it has been 
demonstrated that even with deliberate and concerted effort and active student 
involvement with data the former aim is very difficult to accomplish, and without such 
effort extremely unlikely (Garfield, 1992; Shaughnessy, 1992). In mathematics 
counseling, however, I hoped to have opportunities to address mathematical and 
statistical misconceptions, so I felt that a pre- and post- statistical reasoning assessment 
might reveal changes related to that. Joan Garfield's 20-item multiple choice Statistical 
Reasoning Assessment is well constructed and investigates such faulty heuristics as 
representativeness (e.g., items 9, 1 1, 14), the gambler's fallacy (e.g., item 10), base-rate 
fallacy (e.g., item 12), and correlation as causality (e.g., item 16). 

I gave this assessment as a pre-test at the beginning and a posttest at the end of the 
course. The Statistical Reasoning Assessment was used with permission its author Joan 
Garfield (1998) for purposes of research. See also chapter 8 for discussion of usefulness 
of this instrument in this study. 






Tlic following pages consist of multiple-choice questions about probability and statistics. Read 
tlie question carefully before selecting an answer. 



1. A small object was weighed on the same scale separately by nine students in a science class. 
Tiie weights (in grains) recorded by each student are shown below. 

6.2 6,0 6.0 15.3 6.1 6.3 6.2 6.15 6.2 



Tlie students want to determine as accurately as they can the actual wcigiit of this object. Of 
the following methods, which would you rcconiniend they use? 

. a. Use the most conunon number, which is 6.2. 

b. Use the 6, 15 since it is the most accurate weighing. 

c. Add up tlic 9 numbers and divide by 9. 

d. Throw out tJie 15.3, add up the other 8 numbers and divide by 8. 



2. Tlie following message is printed on a bottle of prescription medication: 

WARNING: For applications to skin areas 
there is a 15% chance of developing a rash. If a 
rash develops, consult your physician. 

Wliich of tlie following is tlie best interpretation of this warning? 

a. Don't use tJic medication on your skin — tliere's a good chance of developing a rash. 

b. For application to the skin, apply only 15% of the recommended dose. 

c. If a rash develops, it will probably involve only 15% of the skin. 

d. About 15 of 100 people who use tiiis medication develop a rash. 

e. There is hardly a chance of getting a rash using this medication. 



3. Tlic Springfield Meteorological Center wanted to dctcmiine the accuracy of tlieir weatlicr 
forecasts. They searched their records for those days when tlie forecaster had reported a 70% 
chance of rain. They compared tliese forecasts to records of whether or not it actually rained 
on those particular days. 



Tlie forecast of 70% chance of rain can be considered very accurate if it rained on: 

a. 95% - 100% of Uiose days. 

b. 85% - 94% of those days. 

c. 75% - 84% of tliosc days. 

d. 65% - 74% of tliose days. 

c. 55% - 64% of those days. 



4. A teacher wants to change tlie seating arrangement in her class in tlie hope tliat it will increase 
tlie number of comments her students make. She first decides to see how many comments 
students make with the current seating arrangement. A record of tlie number of comments 
made by her 8 students during one class period is shown below. 











Student Initials 










A.A. 


R.F. 


A.G. 


J.G. C.K. 


N.K. 


J.L. 


A.W. 


Number of 
comments 





5 


2 


22 3 


2 


1 


2 



She wants to summarize tliis data by computing tlic typical number of comments made that 
day. Of the following methods, which would you recommend she use? 

a. Use tlie most common number, which is 2. 

b. Add up the 8 numbers and divide by 8. 

____ c. Tlirow out die 22, add up the other 7 numbers and divide by 7. 

d. Tlirow out the 0, add up tlie other 7 numbers and divide by 7. 



5. A new medication is being tested to determine its efTectiveness in tlie treatment of eczema, an 

inflammatory condition of the skin. Tliirty patients with eczema were selected to participate in the 
study. Tlie patients were randomly divided into two groups. Twenty patients in an experimental group 
received the medication, while ten patients in a control group received no medication. The results after 
two months are shown below. 



Experimental group (Medication) 

Improved 8 

No Improvement 12 



Control group (No Medication) 

Improved 2 

No Improvement 8 



Based on the data, I think the medication was: 
1 . somewhat effective 



2. basicallv ineffective 



If you cliose option 1. select the one explanation 
below tliat best describes your reasoning. 



a. .40% of the people (8/20) in the 

experimental group improved. 

b. 8 people improved in tlie experimental 

group while only 2 improved in the 
control group. 

__ c. In the experimental group, the number of 
people who improved is only 4 less tlian 
tlie number who didn't improve (12-8), 
while in the control group tlie difference is 
6 (8-2). 

_ d. 40% of the patients in tlie experimental 
group improved (8/20), while only 20% 
improved in the control group (2/10). 



If you chose optios 2. select the one explanation 
below that best describes your reasoning. 



a. In the control group, 2 people improved 

even witliout the medication. 

b. In the experimental group, more people 

didn't get better than did (12 vs 8). 

c. Tlie difference between tlie numbers who 

improved and didn't improve is about tlic 
same in each group (4 vs 6). 

d. In tlie experimental group, only 40% of the 

patients improved (8/20). 



6. Listed below are several possible reasons one might question the results of tlie experiment 
described above. Place a check by every reason you agree with. 

a. It's not legitimate to compare the two groups because there are different numbers of 

patients in each group. 

b. The sample of 30 is too small to permit drawing conclusions. 

c. Tlie patients should not have been randomly put into groups, because the most severe 

cases may have just by chance ended up in one of the groups. 

d. I'm not given enough information about how doctors decided whether or not patients 

improved. Doctors may have been biased in their judgments. 

e. I don't agree with any of these statements. 

7. A marketing research company was asked to detemiine how much money teenagers (ages 13 - 

19) spend on recorded music (cassette tapes, CDs and records). The company randomly 
selected 80 malls located around the country. A field researcher stood in a central location in 
tlie mall and asked passers-by who appeared to be the appropriate age to fill out a 
questionnaire. A total of 2,050 questionnaires were completed by teenagers. On tlie basis of 
tliis survey, the research company reported that tlie average teenager in tliis country spends 
$155 each year on recorded music. 

Listed below are several statements concerning tliis survey. Place a check by every statement 
tliat you agree with. 

. a. Tlie average is based on teenagers' estimates of what they spend and tlierefore could 

be quite different from what teenagers actually spend. 

b. They should have done the survey at more than 80 malls if tliey wanted an average 

based on teenagers throughout the country. 

c. The sample of 2,050 teenagers is too small to permit drawing conclusions about the 

entire country. 

d. They should have asked teenagers coming out of music stores. 

e. The average could be a poor estimate of the spending of all teenagers given that 

teenagers were not randomly chosen to fill out tlie questionnaire. 

f. The average could be a poor estimate of the spending of all teenagers given tliat only 

teenagers in malls were sampled. 

g. Calculating an average in this case is inappropriate since tliere is a lot of variation in 

how much teenagers spend. 

li. I don't agree witli any of tliese statements. 



8. Two containers, labeled A and B, are filled with red and blue marbles in the following quantities; 



Container Red Blue 

A 6 4 

B 60 40 



Each container is shaken vigorously. After choosing one of the containers, you will reach in 
and, witliout looking, draw out a marble. If the marble is blue, you win $50. Which container 
gives you the best chance of drawing a blue marble? 

a. Container A (willi 6 red and 4 blue) 

b. Container B (witli 60 red and 40 blue) 

c. Equal chances from each container 



9. Which of the following sequences is most likely to result from flipping a fair coin 5 times? 

a. H H H T T 

b. T H H T H 

C. T H T T T 

d. H T H T H 

e. All four sequences are equally likely 

10. Select one or more explanations for tlie answer you gave for tlie item above. 

a. Since tlie coin is fair, you ought to get roughly equal numbers of heads and tails. 

b. Since coin flipping is random, die coin ought to alternate frequently between 

landing heads and tails. 

c. Any of the sequences could occur. 

d. If you repeatedly flipped a coin five times, each of these sequences would occur 

about as often as any other sequence. 

____ e. If you get a couple of heads in a row, the probability of a tails on the next flip 
increases. 

f Every sequence of five flips lias exactly the same probability of occurring. 



11. Listed below are tlic same sequences of Hs and Ts tliat were listed in Item 8. Which of tlie 
sequences is least hkely to result from flipping a fair coin 5 times? 

a. H H H T T 

b. T H H T H 

C. T H T T T 

d. H T H T H 

e. All four sequences are equally unlikely 



12. The Caldwells want to buy a new car, and tliey have narrowed their choices to a Buick or a 
Oldsniobile. Tliey first consulted an issue of Consumer Reports, which compared rates of 
repairs for various cars. Records of repairs done on 400 cars of each type showed somewhat 
fewer mechanical problems with the Buick than witli the Oldsmobile. 

The Caldwells then talked to three friends, two Oldsmobile owners, and one former Buick 
owner. Both Oldsmobile ovmers reported having a few mechanical problems, but nothing 
major. Tlie Buick owner, however, exploded when asked how he liked his car: 

First, tlie fiiel injection went out — $250 bucks. Next, I started having 
trouble with tlie rear end and had to replace it. I finally decided to sell it after 
tlie transmission went. I'd never buy another Buick. 

Tlie Caldwells want to buy tlie car tliat is less likely to require major repair work. Given 
what they currently know, which car would you recommend that they buy? 

a. I would recommend that they buy the Oldsmobile, primarily because of all the 

trouble their friend had witli his Buick. Since tliey haven't heard similar horror 
stories about the Oldsmobile, they should go with it. 

b. I would recommend that they buy the Buick in spite of their friend's bad 

experience. That is just one case, wliile tlie information reported in Consumer 
Reports is based on many cases. And according to that data, tlie Buick is 
somewliat less likely to require repairs. 

c. I would tell them that it didn't matter which car they bought. Even though one of 

the models might be more likely than the other to require repairs, they could still, 
just by chance, get stuck with a particular car that would need a lot of repairs. 
Tliey may as well toss a coin to decide. 



13. Five faces of a fair die arc painted black, and one face is painted white. Tiic die is rolled six 
times. Wliich of tlie following results is more likely? 

a. Black side up on five of tJie rolls; white side up on the other roll 

b. Black side up on all six rolls 

c. a and b are equally likely 



14. Half of all newborns are girls and half are boys. Hospital A records an average of 50 birtlis a 
day. Hospital B records an average of 10 births a day. On a particular day, wliich hospital is 
more likely to record 80% or more female births? 

a. Hospital A (with 50 births a day) 

b. Hospital B (with 10 births a day) 

c. The two hospitals are equally likely to record such aii event. 



■^ 



30 40 50 60 70 80 90 100 

Test Scores: No- Sleep Group 



30 40 50 60 70 80 90 100 

Test Scores: Sleep Group 



Examine tlie two graphs carefully. Then choose from the 6 possible conclusions listed below 
tlie one you most agree with. 



a. The no-sleep group did better because none of these students scored below 40 and 
the highest score was achieved by a student in this group. 



b. Tlie no-sleep group did better because its average appears to be a little higlier than 

the average of tlie sleep group. 

c. Tliere is no difference between the two groups because there is considerable 

overlap in the scores of tlie two groups. 

d. There is no difference between the two groups because the difference between their 

averages is small compared to the amount of variation in the scores. 

e. The sleep group did better because more students in this group scored 80 or above. 

f. The sleep group did better because its average appears to be a httle higher than the 

average of tlie no-sleep group. 



\' 



15. Forty college students participated in a study of tlic effect of sleep on test scores. Twenty of i | 

tlie students volunteered to stay up all night studying the night before the test (no-sleep ■ ' 
group). Tlie otlier 20 students (the control group) went to bed by 1 1 :00 p.m. on tlie evening 

before tlie test. Tlie test scores for each group are shown in the graphs below. Each dot on ; ' : 

tiie graph represents a particular student's score. For example, the two dots above the 80 in [ , 
tlie bottom graph indicate Uiat two students in the sleep group scored 80 on the test. 



16. For one month, 500 elementary students kept a daily record of the hours they spent watching 
television. The average number of hours per week spent watching television was 28. The 
researchers conducting tlie study also obtained report cards for each of the students. They 
found that the students who did well in school spent less time watching television than those 
students who did poorly. 

Listed below are several possible statements concerning the results of this research. Place a 
check by every statement that you agree with. 

a. The sample of 500 is too smallto permit drawing conclusions. 



b. If a student decreased the amount of time spent watching television, his or her 
performance in school would improve. 

c. Even though students who did well watched less television, this doesn't necessarily 

mean that watching television hurts school performance. 

d. One month is not a long enough period of time to estimate how many hours the 

students really spend watching television. 

e. The research demonstrates that watching television causes poorer performance in 

school. 

f I don't agree witli any of these statements. 



17. The school committee of a small town wanted to determine the average number of children per 
household in their town. They divided tlie total number of children in the town by 50, the total 
number of households. Which of the following statements must be tnie if the average children 
per household is 2.2? 

a. Half the households in the town have more than 2 children. 

b. More households in the town have 3 children than have 2 children. 

c. There are a total of 1 10 children in the town. 

d. There are 2.2 children in the town for every adult. 

e. The most common number of children in a household is 2. 

f None of the above. 



18. When two dice are simulataneously tliiown it is possible tliat one of the following two results 
occurs: 

Result J: A 5 and a 6 are obtained. 

Result 2: A 5 is obtained twice. '' 

Select tlie response tliat you agree with tlie most: 

a. Tlie chances of obtaining each of tlicse results is equal 

b. Tliere is more chance of obtaining result 1 . 

c. Tliere is more chance of obtaining result 2. : 

d. it is imposible to give an answer. (Please explain why) 

19. When three dice are simultaneously thrown, which of the following results is MOST LIKELY ;! 
to be obtained? Ji 

a.;toM//V:"A5, a3anda6" ^1 

b. Result 2: "A 5 three times" ; 

c. Result 3: A 5 twice and a 3" 

d. All tliree results are equally likely 

20. When three dice are simultaneously thrown, which of these three results is LEAST LIKELY to 
be obtained? 

a. Result I: "A 5, a. 3 and a 6" 

b. Result 2: "A 5 three times" 

c. Result 3: A 5 twice and a 3" 

d. All three results are equally unlikely 



432 

5. The Algebra Test, adapted from the Chelsea Diagnostic Algebra Test. This test is one 
often designed as a diagnostic instrument to be used "both for ascertaining a child's 
[aged 12 through 15+ years] level of understanding and to identify the incidence of 
errors" by the mathematics research team of the British Social Science Research Council 
Program 'Concepts in Secondary Mathematics and Science' (CSMS)(Brown, Hart, & 
Kuchemann, 1 985). The research was carried out "broadly within a Piagetian 
framework." In particular, the algebra test specifies four levels of understanding of the 
algebraic from level 1 at which a letter can be evaluated by recalling an arithmetical 
relationship and letter objects to be collected are all of one type, through level 4 at which 
the letter is understood at least as specific unknowns or generalized numbers (and in 
some cases as variables) and two operations can be coordinated. Sokolowski designed a 
fifth level at which the letter is understood as having "a range of numbers (a dynamic 
view) that is, as a true variable and coordinated operations can be reordered and 
reconfigured" (Sokolowski, 1997, pp. 97-98). Sokolowski's level 5 items have not been 
subjected to the rigorous clustering and leveling analysis applied by CSMS to the level 1 
through 4 questions, however. Sokolowski also made minor language and setting 
changes in the test to make it comprehensible to students in the New England area of the 
U.S. 

In this study it was expected that some participants' difficulties with the 
mathematics could be linked directly to weak mathematical backgrounds, gaps, and 
primitive understanding of the algebraic variable. Others' difficulties were expected to be 
in spite q/" sound mathematical and algebraic concepts. I beheved that the Algebra Test 
would be a valuable tool for helping pinpoint a symptomatic (mathematical) focus for the 



433 

former, and an explanation (removable by education) other than intrinsic inability for 
their troubles. For the latter it could be used as evidence to reflite their negative opinions 
of their mathematical functioning that were contributing to their helplessness and poor 
achievement. I used the Algebra Test with permission. See chapter 5 and 6 for further 
discussion. The Algebra Test was used with permission for the purposes of this research. 
See chapter 6, 7, and 8 for further discussion. 



^^"^ - T>«che 



Algebra Test 



Practice Item I 

1 . What number does a + 4 stand for if a = 2 



- 






ifa = 5_ 






Practice Item 2 






2. Fillin the blanks: 






Work down the page 






X-* 3x X-* x^3 


X -* 7x 


X —*■ 


2-> 6 5—8 


2 — ► 


3 -> 


5 -> 4 -> 






n —*■ 





x + S 



1 . Fill in the blanks: .t — * .r + 2 x —* 4x 

6 ->■ 3 -> _ 



2. Write the smallest and the largest of these: smallest largest 
n + 1, « + 4, n-3, n, - «-7 



3 . Which is larger, 2n or /i + 2? 
Explain. 



4. 4 udded to n can be written as « +4. n multiplied by 4 can be written as 4/i . 
Add 4 to each of these: Multiply each of these by 4: 

8 n + 5 3/1 8 /i + 5 3« 



5. Ua^b =43 If n -246 = 762 If<?+/ =8 

a+b+2 = 71-247 = e+f+g=_ 



6. What can you say about a if a + 5 = 8 

What can you say about b if h + 2 is equal to 2h 



1 . What are the areas of these shapes? 



A = 



A = 



10 



A = 




c 2 



A = 





8 . The perimeter of this shape is equal 
to 6 + 3+ 4 + 2, which equals 15. 



What is the 

perimeter of this shape? P = . 



9. This square has sides of length ^. 



So, for its perimeter, we can write P = 4^. 



What can we write for the perimeter of 
each of these shapes? 




P = 



\s 5 




P = 



Piut of this ligun: is nol 
dnwn. Tliere arc n sides 
altogelher, all of length 2. 

P = 



10. Small apples cost 8 cents each and small pears cost 6 cents each. 



If a stands for the number of apples bought 
and p stands for the number of pears bought, 
what does 8a + 6/7 stand for? 



What is the total number of fhiits bought? . 



1 1 . What can you say about u if « = v + 3 

and V = 1 



What can you say about m if m = 3/? + 1 

and rt = 4 _ 



12. If John has y compact discs and Peter has P compact discs, what could 
you write for the number of compact discs they have altogether? 



13. a + ia can be written more simply as 4a. 
Write these more simply, where possible: 
2a + 5a = 

3a-{b + a)= . 

a + 4 +a-4 = . 

3a -b + a = 



la 


^5b = 


{a + b) + a = 


2a 


^5b +a = 


(a- 


-b) + b = 



ia + b) + {a-b) = . 



14. What can you say about r if r = ^ + r 

andr + i+r=30? 



15. In a shape like this you can determine the number of diagonals from one vertex 



by taking away 3 from the number of sides. 
So, a shape with 5 sides has 2 diagonals; 

a shape with 57 sides has diagonals; 

a shape with A: sides has diagonals. 

16. What can you say about c if c + J = 10 

and c is less than d 




17. Mary's basic wage is $200 per week. 

She is also paid another $7 for each hour of overtime that she works. 

If h stands for the number of hoiu-s of overtime that she works, and 

if W stands for her total weekly wages (in $), 

write an equation connecting W and h.. 



What would Mary's total weekly wages be if she 
worked 4 hours of overtime? 



18. When are the following true - always, never, or sometimes? 

Underline the correct answer: 

A + B + C = C + A + B Always Never Sometimes, when . 

L + M + N = L + P + N Always Never Sometjmesi when . 



19. a = h + 3. What happens to a if * is increased by 2? , 
/ = 3^ + I What happens to / if g is increased by 2? . 



20. Bagels cost b cents each and muffins cost rn cents each. 
If 1 buy .4 bagels and 3 muffins, 
what does 4b + 3/n stand for? 



21. If this equation (x + 1)3 + jc = 349 is true when a: = 6, 
then 
what value of .r will make this equation , {5x + 1)^ + 5x = 349 , true? 



x = 



22. Fine point black pens cost $3 each and medium point red pens cost $2 each. 

I went to Staples in Salem, New Hampshire, and bought some of each type of pen, 
spending a total of $25. 

If & is the number of black pens 

and if r is the number of red pens bought, 

what can you write about b and r? 



23. You can feed any number into this machine: 
Can you find another machine that 



+ 10 


—^ 


X 5 



has the same overaill effect? 



X — w~ + 



Note: The Chelsea Diagnostic Algebra Test (Brown et al., 1985) was used for this research with the written 
permission of its publishers. 



Name_ 
Date 



Algebra 
Levels of Understanding 

Course/Semester 



Last Math Course 



Level 1 


Level 2 


Level 3 


Level 4 


i-^v'el ^ 


5(a) 


7(c) 


4(c) 


3 


I'^CL 


6(a) 


9(b) 


5(c) 


4(e) 


as 


7(b) 


9(c) 


9(d) 


7(d) 


8 


11(a) 


13(b) 


13(e) 




9(a) 


11(b) 


13(h) 


17(a) 




13(a) 


13(d) 


14 


18(b) 






15(a) 


15(b) 


20 








16 


21 










22 




4/6 


5/7 


5/8 


6/9 





Totals 



1: 



3: 



Math Course Taking History 
High School 

Freshman 

Sophomore 

Junior^ 

Senior 



College 

Freshman _ 
Sophomore 

Junior 

Senior 



442 



6. Observation Tools: 

a. Music Staff Class Interaction Analysis Chart 
Start Time: End time: 



l:S" 



i;Q 



_S^ 



S:S 



Start Time: End time: 



I:S 



1:Q 



S:Q 



.iS. 



Start Time: End 

time: 



I:S 



I:Q 



S:Q 



S:S 



TS indicates Instructor's Statement; I:Q indicates Instructor's Questions; S:S indicates Student's 
Statement; S:Q indicates Student's Questions 



6. b. Class Layout Observation Form 



Front Right 



443 




Front Left 



444 



6. c. Problem Working Session Interaction Chart: With seating for Class 13 


Time 




pa 




A 




R 




B 




Jillian 




Ka 




C 




M 




J 




L 




M 





Note. A = Autumn; B = Brad; J = Jamie; Ka = Karen; Ke = Kelly; L = Lee; Mi = Mitch; Mu = Mulder; P = 

Pierre; R = Robin. 

^Seated with Pierre front right going counterclockwise to Mitch seated front left. 



445 



6. d. 


Class # 


Summary 




Course/Semester 






Professor 


Date 






Location 



Classroom configuration and individual's locations: 



Summary of class 



Interactions 



Teaching/Learning 



Participant/observer issues 



Thoughts for next class: 

©Jillian M. Knowles, Lesley University, Summer 2000 



446 



Appendix D 

Research Information and Informed Consent Forms 

Learning Assistance for a College Undergraduate Mathematics Class 

Doctoral Dissertation Research 

Jillian Knowles Summer, 2000 

My aim in this research is to investigate the role of a number of different types of 
learning assistance interventions in helping students who are taking a required 
undergraduate mathematics course to not only pass the course, but also to improve their 
grasp of and approach to mathematics. In order to do this, in all aspects of the research, I 
will be investigating each participating student's own ideas and feelings on his or her 
issues around mathematics learning at college. 
I, as the participant researcher, will be: 

• attending and observing all the Summer 2000 PSYC/STAT 104 classes. This will 
include my giving the class two pre and post surveys on beliefs and feelings around 
mathematics. [Complete confidentiality is assured.] 

• I will be organizing and observing (including audio-taping these sessions) a weekly 
study group that will meet before class at 4:30pm on Wednesdays m University 
Center, Room 254 [Complete confidentiality is assured.] 

• I will be offering Drop-In and by-appointment mathematics tutoring in the Learning 
Assistance Center, Greenville campus, Room203. [Complete confidentiality is 
assured], and, finally, 

• I will be offering one-on-one mathematics counseling (audio-taped, transcribed and 
analyzed) to volunteers who want to work on their emotional and mathematical 
background issues in order to improve their approach to and achievement in 
mathematics in life and in college. There is a possibility of follow-up of individuals 
per mutual agreement with me. [Complete confidentiality is assured.] 

Please Note: 

1 . The personal identity of each participant in this study will be kept confidential. Each 
participant will be assigned an assumed name (You can choose!). All analysis and 
reporting will use these assumed names and the setting will be disguised. 



447 



2. Your participation or non-participation in this study will in no way affect your 
grade in this course. If you do not wish to participate you will indicate that by not 
filling in the class surveys and by not signing the permission sheet at the study-group 
and Drop-In. Alternately, on the class surveys, you may be willing to complete them 
using a number rather than your name, remembering the same number for the post 
tests. In that way, there will be complete data for the class but your individual 
responses will not be directly linked to you. 

3. You are encouraged to take advantage of any or all of the above learnmg assistance 
offerings. Being involved in one does NOT mean you cannot take advantage of 
others. 

Analysis will involve some quantitative and much qualitative work. Quantitative analysis 
of the pre and post surveys using, amongst other tests. Student's t test difference of 
means for dependent groups wall be clarified using qualitative data. Qualitative analysis 
will involve developing grounded theory. This means that I will have to be continually 
noting and setting aside my own assumptions about what are your key issues around your 
mathematics learning and listening to and hearing you. I vWll work at producing draft 
theories for you to look at and critique, until a grounded theory is developed. This study 
will then be reported in my doctoral dissertation for Lesley College, Cambridge MA. 

Jillian Knowles, 

Local Identification and Contact Information 



448 
A Call for VOLUNTEERS 

I'm looking for people who want to learn how to do their mathematics more effectively. 
I need several volunteers who will agree to meet with me regularly (for 1 hour per week 
or once every other week for 1 hour per session) for the duration of the Statistics in 
Psychology PSYC/STAT 104, Summer 2000 course, to engage in one-on-one 
mathematics counseling — working on both your mathematics and also your emotional 
issues around mathematics. 

If you have issues around mathematics learning that you feel may make it harder to 
succeed in this course, maybe this could help. I have worked with college students, 
teaching, tutoring, and helping them with their mathematics for many years. In my 
doctoral studies I have been looking for better, more effective ways to do this. In this 
dissertation research project, I wish to explore these new ways with students who want to 
improve how they do mathematics. It will be completely confidential and should lead to 
improved ways of doing mathematics. 

If you would like to work with me, please respond "Yes" on the attached index card 
which I will collect with your surveys. If you want more time to think about it, come to 
see me at the Learning Assistance Center, Room 203, Greenville campus, call me at the 

Learning Assistance Center at or at home or e-mail me at ^or at . 

I DO need to know by Wednesday, June 7, because the course time is so short, so you 
could let me know at the study group or m class on June 7. 



Jillian Knowles 
Local Contact Information 
Doctoral student 
Lesley College, 
Cambridge, MA 



449 



DISSERTATION RESEARCH INFORMED CONSENT FORM 

I, , a) affirm that I have read and Jillian Knowles has 

explained the objectives of her research, the procedures to be followed and the potential risks and 
benefits. yes/no 

b) understand that my participation or 
nonparticipation in this research project will not affect my grade in Dr Paglia's Statistics in 
Psychology PSYC 402 Summer 2000 class yes/no 

c) understand that I am free NOT to respond to 
any part of the research yes/no 

d) understand that I can withdraw from the 
research at any time yes/no 

e) affirm that I have volunteered to be involved 
in this research of my own free will, without coercion by Jillian Knowles or any other person 

yes/no 

f) agree that the information I give may be 
discussed only with Jillian Knowles' dissertation committee members at Lesley College, 
Cambridge, MA, using my name/under an assumed name, and used to write her dissertation for 
her doctoral degree. Otherwise all materials and information about me she gathers will be 
kept completely confidential — in particular, they will NOT be shared with any persons or 
institutions within the University of New Hampshire at Manchester 

yes/no 

g) assert that if Jillian Knowles chooses at some 
time to include any information I give in a published article/book, she may do so with/without my 
written/verbal consent yes/no 

h) Jillian Knowles will not publish materials 
about me without having allowed me to review the relevant part of article/book first yes/no 

i) Jillian Knowles will keep audio-tapes and 

transcripts of this data in a secure place and will only allow direct access to it by her dissertation 

committee. Access by others will only be allowed with my verbal/written permission yes/no 

Signed by me this day of , 

20 

Name 

Address 

Phone e-mail 



450 

Individual Mathematics Counseling 

Sign-Up Card 

A personalized copy of this 4inch by 6inch response card was given to each student in the 

PSYC/STAT 104 class during the second class of the course. All students responded and 

returned their cards at that time. 



Student Name 6/5/00 

I would like to meet with Jillian Knowles for 

I I 1 hour per week 

1 hour every other week 

[Please check one] 

beginning this week (if possible) until the end of the summer 
2000, PSYC/STAT 104 Statistics course, to do one-on-one 
mathematics counseling. [Please 

Yes/No circle 
Signed by (optional) one] 



451 
Appendix E 

Coding and Analysis 

In this study looked at students' sense of mathematics self, their mathematics 
internalized presences, and their mathematics attachments to better understand their state of 
mathematics functioning or mathematics mental health that would lead to strategic 
approaches to helping them negotiate their college mathematics course, I analyzed our 
interactions, their behaviors and utterances in class, study group, and in counseling sessions, 
their responses to the instruments and their mathematical products in terms of these three 
dimensions. I wished to determine if the three dimensions provided a reasonable framework 
for understanding their mathematics functioning but also if there were important elements 
that could not be understood this way. I wanted to see if students' affective and cognitive 
symptoms of dysfunction could be better understood via this framework. 

The central task for the relational mathematics counselor in this study was 
continually culling relevant data from the voluminous observations and then processing the 
data in order to help the student grow in his mathematical functioning and relationships. 
That processing as Arlow (1995) and others in the psychoanalytic tradition point out has "an 
aesthetic [aspect] that depends on empathy, intuition, and introspection" (p. 144) and a 
cognitive aspect that "depends on rationally assembled, methodologically disciplined 
conclusions from the data of observation" (p. 44). Since in psychoanalysis as in mathematics 
counselmg, life and class events change the context and meaning of observations, the many 
variables are impossible to control; hence the need to limit the dimensions of the issues 
under investigation in any empirical investigation (Arlow, 1995). The stance that 
psychoanalysis and empirical investigation are antithetical is giving way to more and more 
nuanced standardized methodologies such as using guided central relationship measures to 
guide the therapy more systematically and allow for more empirical evaluation of 



452 
techniques, their underlying rationales, and outcomes (Luborsky & Luborsky, 1995). For my 

purposes here the use of a modified guided central mathematics relationship measure to 

guide the counseling and, via analysis, to trace its path retrospectively, seems appropriate. 

My task in tracing the path of mathematics counseling and analyzing its efficacy is 
in some senses easier than the task of the psychoanalyst. Since the central symptomatic 
focus for each tutee is mastering the mathematics course, his mathematical behaviors in the 
classroom (see Table E2 and Table E3) and in the counseling sessions and his mathematical 
products for the course: homework, projects and especially exams(see Table E4), provide 
central data for charting his progress. I was also able to follow targeted mathematics 
affective symptoms and their changes through pre and post feeling and belief/attitude 
surveys. It was the relational changes that I hypothesize underlie his mathematical cognitive 
and affective changes that I need a guided central relationship measure to gauge (see Table 
El). In this also I have an advantage over the psychoanalyst, who only sees the client in the 
counseling setting, since I see the student not only in the counseling setting but also in the 
central forum of his present mathematics life — the classroom^ — so what he reports in the 
counseling session of his experiences in class I and the instructor also observe (see Table 
El). 

On the other hand the major disadvantage in trying to formulate a student's central 
relational pattern or conflict lies in the fact that the central focus in the mathematics 
counseling is on the student doing mathematics rather than on his relational, albeit 
mathematics relational, conflicts. This means that relatively little time is spent in a 
mathematics counseling session in talking about his past and present mathematics 
relationships. Therefore there is substantially less direct student-initiated relational data from 
the sessions, especially relational data with respect to the counselor. This may be related to 
the predominance, from the student's perspective, of the tutor role over the coimselor role in 



453 
this setting and his concomitant expectations of and desires for what mathematics 

counseling would entail — that is, mathematics tutoring. 

The basic organizational unit I used to do identify a student's central mathematics 

relational pattern was the relational episode. This involved first locating and identifying 

narratives' (called relationship episodes) and then reviewing the relationship episodes and 

extracting the central relationship theme from them (Luborsky & Luborsky, 1 995). Three 

components that Luborsky (1976) finds prominent in these relational episodes are: what the 

patient wanted from other people; how the other people reacted; and how the patient reacted 

to their reaction. Other researchers include disguised allusions, acts of self, expectations of 

others, consequent acts of others towards self and consequent acts of self towards self (cf 

Gill and Hof&nann, 1982; Schacht et al., 1984) 

My adaptations for mathematics achievement settmg are: Central relationship pattern 

with respect to self: 

1 . what student wants/expects from self; 

2. student's achievements; 

3. how other people reacted to student's achievements; 

4. how the student has reacted to others' reactions to his achievements; and 

5. mathematicsaspart of self 

Central Relationship Pattern with respect to internalized presences: 

1 . what student wanted/expected from other people; 

2. how other people reacted to student; 

3. how the student reacted to their reactions; and 

4. mathematics as internalized other 



454 
Central (interpersonal) Attachment Pattern: 

1 . what student wants/expects from other people; 

2. how other people react to student; 

3. how the student reacts to their reactions; and 

4. attachment to mathematics 

A crucial concern in understanding a student's central relationship pattern from a 
relational conflict perspective is the understanding that the student is dealing with patterns of 
conflict, parts of which he is conscious and parts of which he is unconscious. This means 
that his verbal statements and behaviors will likely include ones that appear to and some that 
do contradict his basic wishes. In order to identify a student's central relationship pattern 
given this difficulty I adopted the following four principles developed by Luborsky and 
Luborsky(1995): 

1 . The central conflictual relationship theme may have an opposite conflicting less 
conscious theme, 

2. A wish frequently expressed may have a less frequent (but perhaps more intense) 
version of that wish in reduced awareness, 

3. Instances of denial are likely to point to content that is in reduced awareness, 

4. If a student refers to a history of difficulties with awareness this might infer present 
similar difficulties that he does not acknowledge, (p. 345) 



455 



Table El 



Analysis scheme for Counseling Session Data: Student's Mathematical Relationality 




Mathematics Self 


Mathematics 


Mathematics Interpersonal 






Internalized Presences 


Environment 


Relational 


Mathematics Identity 


Object relations 


Attachments 


Assessment 


Central Relationship Patterns 


Cenfral Relationship 


Central Relationship 


Categories 


1. With self 


Patterns 


Patterns 




2. With mathematics 


1 . With internalized 


1 . With others now 






others 


2. With mathematics now 






2. With mathematics 





Central what a student wants/expects 
relationship from self; a student's 
measure achievements; how other people 
categories reacted to student's 

achievements; how the student 
reacted to others' reactions to his 
achievements; and mathematics 
as part of self 
Metaphor Survey" 



what student 

wanted/expected from other 
people; how other people 
reacted to student; how the 
student reacted to their 
reactions; and 
mathematics as internalized 
other 
Metaphor Survey 



what a student's 
wants/expects from other 
people;how other people 
react to student; how the 
student reacts to their 
reactions; and attachment to 
mathematics 
Metaphor Survey 



Mathematics Testing or mathematics anxiety 
Affect as extinction anxiety: History (re 
teacher 's mirroring and 
invitation to idealize), Feelings 
Survey, Metaphor Survey; Test 
Taking behaviors. Mathematics 
depression, learned helplessness 
as empty depression; 
mathematics grandiosity 
History (mirroring and 
invitation to idealize). Beliefs 
Survey, JMK Affect Scales; class 
and counseling behaviors 



Testing or mathematics 
anxiety as social anxiety, 
adjustment disorder, PSTD, 
phobia, ... 

Feelings Survey, Metaphor 
Survey, History (re critical 
incidents); classroom 
behaviors versus classroom 
" reality. " Mathematics 
depression related to a 
severe mathematics super 
ego/ internal saboteur 



Testing or mathematics 

anxiety as separation (from 

teacher or mathematics) 

anxiety 

History (re separation, loss, 

change) 

Feelings Survey Metaphor 

Survey; teacher and 

counselor related behaviors. 

Mathematics depression 

related to separation or loss 



Mathematics Elementary Mathematics: 
Cognition PSYC/STA T 1 04 Exams, 
Arithmetic for Statistics 
Assessment; High School 
Mathematics: HS courses/ 
grades. The Algebra Test; Intro 
Statistics: PSYC/STAT 104 
Exams 



Elementary Mathematics: 
PSYC/STA T 104 Exams, 
Arithmetic for Statistics 
Assessment; High School 
Mathematics; HS courses/ 
grades. The Algebra Test; 
Intro Statistics: PSYC/STAT 
104 Exams 



Elementary Mathematics: 
PSYC/STA T 104 Exams, 
Arithmetic for Statistics 
Assessment; High School 
Mathematics: HS courses/ 
grades. The Algebra Test; 
Intro Statistics: PSYC/STAT 
104 Exams 



Counselor's 



counfertransference 



" The items in italics are instruments, protocols, mathematics products, demographic and behavioral data 
that were used in conjunction with audiotaped counseling session data to develop a profile of a student's 
mathematics fimctioning and his central mathematics relational pattern. 



456 



Table E2 



Analysis of Lecture Session Student Exchanges with Instructor 



Student Questions 


Student Answers 


Student Comments 


Timing^ 


Timing 


Timing 


Relevance 


Accuracy 


Relevance 


Topic: 


Topic: 


Topic: 


1. current content: 


1. current content: 


1. current content: 


mathematics; 


mathematics; 


mathematics; 


application; 


application; 


application; 


personal 


personal 


personal 


2. course strategy 


2. course strategy 


2. course strategy 


3. grading 


3. grading 


3. grading 


Level of certainty: 


Level of certainty: 


Level of certainty: 


1 . affective, 


I . affective, 


1 . affective. 


2. cognitive 


2. cognitive 


2. cognitive 


Frequency 


Frequency 


Frequency 


Development 


Development 


Development 


Implications re student's 


Implications re student's 


Implications re student's 


1. mathematics self 


1 . mathematics self 


1 . mathematics self 


2. internalized 


2. internalized 


2. internalized 


presences 


presences 


presences 


3. attachments: to 


3. attachments: to 


3. attachments: to 


teacher; to 


teacher; to 


teacher; to 


mathematics 


mathematics 


mathematics 


Implications re student's 


Implications re student's 


Implications re student's 


auditory processing 


auditory processing 


auditory processing 


Central relational conflict or theme 



''Timing is judged in terms of the extent to which the student's verbalization is linked in a 
timely manner with the instructor's utterance. For example, on a number of occasions 
Robin answered Ann's question with the correct answer to a. previous question; her 
timing was off. 



Table E3 



457 



Analysis of Student's Problem Working Session Behaviors 



p 




Topic/Task: 


Seated beside: 


Tools: 


Interaction 


Interaction 


A 






1. Left 


1 . text 


with 


with 


R 








2. provided by 


Instructor 


researcher 


A 






2. Right 


instructor 






M 








3. student 






E 








aids: 






T 
E 
R 

S 








calculator, 












notes,... 










Cognitive 


Peer relational 


Mathematics 


Student- 


Student- 






preparation: 


behaviors: 


learning style 


teacher 


researcher 






I . Background 


I. social 


Behaviors: 


relational 


relational 






2. Homework 


learner ' 


I. Analytic 


behaviors: 


behaviors: 








2. voluntary 


(Mathematics 


I. Secure 


1. Secure 








loner" 


Learning 


attachment ■* 


attachment'' 








3. involuntary 


Style I)" 


2. Insecure 


2. Insecure 








loner" 


2. Global 
(Mathematics 
Learning 
Style II)'' 

3. HarmonicI' 

4. Harmonic ir 


avoidant '' 

3. Insecure 
dependent "* 

4. Insecure 
disorganized'^ 


avoidant '^ 

3. Insecure 
dependent '' 

4. Insecure 
disorganized'' 


Changes 






R 

E 


Implications 


Implications 


Implications 


Implications 


Implications 




L 


Re: 


Re: 


Re: 


Re: 


Re: 




T 


1. Self 


1. Self 


1. Self 


1. Self 


1. Self 




O 


2. Internalized 


2. Internalized 


2. Internalized 


2. Internalized 


2. Internalized 




N 
A 


presences 


presences 


presences 


presences 


presences 




L 
I 
T 
Y 


3. Attachments 


3. Attachments 


3. Attachments 


3. Attachments 


3. Attachments 








Central relational conflict 


or theme 





" I designated as social learners in this group Lee, Mulder, and Robin because they always chose to work with 
people beside them if they were willing; I designated as involuntary loners Pierre and Jamie because they 
seemed to be working alone not by choice but because of personal issues; I designated as voluntary loners 
Autumn, Catherine, and Karen because they showed no interest in working with others (except Ann or me). 
Catherine was willing to help someone if he asked (e.g., Mulder) but never asked to check with anyone. ""See 
Davidson, 1983; Witkin et al, 1967 and chapter 2 discussion. '^ Harmonic I balance of Mathematics Learning 
style I & 11 more I; Harmonic II balance of Mathematics Learning style I & II more II. See Krutetskii, 1976, 
and chapter 2. '' See Bowlby, 1973 and chapter 2. 



458 



Table E4 





Protocol for Analysis of Exam Question Solutions 






The 
question 


Pre-Exam: 

Class 
treatment, 

student 
reaction and 
Counseling 
preparation 


Student's 
out of class 
preparation 


Errors 


Trouble- 
shooting 
efforts: 


Instructor 
Grading 


Post-exam 
Counseling 


Defining 

the 
problem: 
concepts 






1 . understanding 

the question 

2. misconceptions 

3. confusions with 

4. other 


1. affective 

2. cognitive 






Planning 

the 

solution: 

procedures 






1 . formula sheet 

2. data-symbol 
linking 

3. choice of 
formula 

4. strategy/ 
layout 


1. affective 

2. cognitive 






Carrying 
out the 
solution: 
algebra 






1 . multiple uses of 
letter symbols 
2. algebra 


1. affective 

2. cognitive 






Carrying 
out the 
solution: 
arithmetic 






1. arithmetic 

2. order of 
operations 


1 . affective 

2. cognitive 






Conclusion 
Checking 
and 
reporting 






1 . reasonableness of 

solution 

2. units 

3. interpretation of 

solution 


1 . affective 

2. cognitive 






Individual 
Patterns 














Class 
Patterns 















' These narratives are extracted not only fi-om direct student reports but also from discussion of classroom 
interactions, metaphor and survey responses and mathematics focused interactions with the counselor. 



459 

Appendix F 
Researcher and Student Seating 

Ann in her final interview (Interview 3) noted that she had not taught a class 
before in which there seemed to be so much change in seating arrangement. She 
wondered if the different physical arrangement of the classroom from the usual rows of 
individual chairs was a factor. In previous classes she had taught students had mostly 
maintained the seating positions they had taken in the first class, changing only to sit in 
seats adjacent to the original. We both also wondered about the effect of my choices of 
seating on the choices by the students. 

Researcher seating. I had struggled with my seating choices throughout the 
course (see Figure FI). In my role of researcher, I wanted to be as much an observer and 
as little a participant as possible. As the class progressed, and I realized that I was 

No. of Which 
Position Times Classes 



□ 
■ 



(Class 14) 
(Class 8) 
(Class 11) 
(Class 17) 
(Classes 6,7,& 12) 
(Classes 1,2,&3) 

(Classes 13, 15,16,18,&. 



j/o 



tjor 



left 



DDHna 



front 
chalkboa 



D 
D 



SX- 



nnannn 

right 



back 



Figure Fl. Jillian's seating positions for the PSYC/STATS 104 course, second floor. 
Riverside Center, Brookwood State University, Summer 2000 



choosing to sit only on the right side, albeit in various positions, I decided that I needed a 
perspective of the class and students from the left. Eventually, towards the end of the 



460 

course, I decided that a better perspective of the whole class might come from the end of 
the table beside Karen (see Figures 1 and 2). 

Student seating. Karen (12 times) and Catherine (10 times) (left back comer, see 
Figure 12) were the most consistent in their choice of seating of the class, although 
Autumn (9 times) mostly right middle and Jamie (9 times) left middle were almost as 
predictable. Brad (right back comer), Robin (right side mostly next to Brad in the right 
back comer), Mitch (left middle to left front), and Lee (mostly left side beside Mitch) 
each had her or his discemable partem (7 to 8 times each). Pierre usually sat close to the 
front evenly on left and right sides — ^presumably to maximize the use of his tape-recorder 
for each lecture. Mulder showed the most inconsistency, perhaps because he was usually 
a few minutes late to class (he had to transport his mother) so he had to find an empty 
space — he was more often on the right or at the back (See Figure F2 for a most 
representative seating arrangement). 



B Mitch [^ Karen Door/ 

B Lee Q Brad 

■ Jamie V^ Robin 

E^ or lH Catherine front chalk boaid 

^m Autumn 



left 



[— 1 or 



Pierre 



DBD 



nnannn 



n 
n 



right 



tack 



461 



Ellen only came to the first class and sat on the left at the front. Kelly dropped 
the class the day of the second test (the 9"" class). She sat towards the front either on the 
left or the right as she tended to be late and there was usually a seat or two unoccupied 
towards the front. Floyd came to only four classes in this room and sat at the back each 
time. This data contrasts somewhat with Arm's perception of constant change in student 
seating. Most students were relatively consistent in their seating choices or patterns. 



462 



Appendix G 
PSYC/STAT 104 Instructor Syllabus and Selected Handouts 



PSYC/<T'AT- Statistics in Psychology 
j 0% 
Summer 2000 
M/W 6:00 -8:20 May 31" - August 2"'' 

Professor: Aia v^ 'Pov-i'-e^i^ Ph.D. 
Office Information: 
Phone: e-mail: 



Summer Office Hours : by appointment 

Due to several advisory responsibilities at and 

my summer schedule is extremely inconsistent. PLEASE 
DO NOT take this to mean I am inaccessible, just that my schedule 
fluctuates from week-to-week. Please, feel free to contact me anytime 
to schedule an appointment. 

Required Text Book : 

Pagano, R.R. (1998). Understanding Statistics in the Behavioral Sciences - Fifth 
Edition . Pacific Grove, CA. Brooks/Cole Publishing Company. 

Course Overview: 

Psyd/^hxf'iO^. - Introduction to Statistics in Psychology will provide a 
comprehensive overview of the basic statistical concepts utilized in psychological 
research. Many, if not all, of these concepts are utilized in other disciplines as 
well. In order to comprehend statistics, it will be necessary to initially learn the 
material at a conceptual level. Calculations and computer modules will be 
required to advance your understanding of the statistical concepts. 

Computers are an essential part of the psychology program at and are 

extensively used in the field psychology. These computer assignments are 
intended to illustrate the ease that computer statistical analysis provides with 
large data sets. Although there will t»e some initial frustration, as you become 
familiar with the computer program itself, you will witness the convenience that 
computers provide to statisticians. Familiarity with Mini-tab, a computer program 
available for statistical analysis, is a university-wide requirement for this course. 
Mini-tab is available on the mainframe computer. This version of Mini-tab is 
somewhat archaic, but will provide the necessary exposure to statistical analysis 
on the computer. 



Course Goals : 

1. To provide a basic overview of the statistical concepts utilized in empirical 
research. 

2. To facilitate a comfortable relationship with statistical concepts. 

3. To increase the conceptual understanding of the various statistical 
analyses utilized in research. 

4. The increase understanding and critical thing about the statistics that the 
media presents. 

5. To increase familiarity with statistical calculations and computer analysis. 
Course Requirements : 

1. You are expected to attend class on a regular basis. You are expected to 
read the text and compute statistical calculations in preparation for class 7 
lectures and the tests. 

2. Tests : There will be a total of 5 tests. Each of the first 4 tests will 
be worth 20% of your final grade. The 5"' test is a conceptual 
comprehensive exam and is worth 10% of your final grade. You will have 
the full class time to complete the exams. All 5 of the combined exam 
grades will determine 90% of your final grade. 

3. Computer Assignments : You will be required to complete one 
computer module independently for 2% of your final grade. Additionally^ 
you will be required to complete an additional computer module with a 
few of your classmates. In addition to completing the module as a group, 
you will be required to present this information to the class. The group 
presentation and paper are worth 8% of your final grade. Handwritten 
papers will not be accepted! Computer Assignments must be typed!!! If 
papers are turned in after the due date, you will loose one letter grade for 
each date that the paper is late. 

The group presentation and paper should include: 

• All members of the group contributing to the oral presentation. 

• A review of the method that the module illustrates. 

• A visual display of the entire statistical analysis. 

• An overview of the "Interpretation" portion of the assignment. 

• A brief written commentary (specific form will be distributed to the 
class) of the efforts of the group (ex. meetings, attendance at 
meetings, designation of tasks, etc.) completed separately by each of 
the group memt)ers. If any member does not contribute to the group 
assignment, it will be reflected on that individual's grade for this 
assignment 



4. Homework : "Questions & Problems" are located at the end of each 

chapter in the text. You are not required to turn in the homework to me, 
but be sure to do these assignments, as they are essential to your 
understanding of the course material. These "Questions and Problems" 
provide an excellent review for the tests. 

UNH Grading Scale: 



Final grades will be based on the following scale: 



B+ = 87% - 89% 
C+ = 77% - 79% 
D+ = 67% - 69% 



A = 93% ■ 


-100% 


A- 


= 90% • 


■ 92070 


B = 83% ■ 


- 86% 


B- 


= 80% • 


- 82% 


C = 73% ■ 


•76% 


C- 


= 70% ■ 


- 72% 


D = 63% 


- 66% 


D- 


= 60% 


- 620/0 


F= 0%- 


S9% 









Course Policies : 

A calculator with a square root key is required for this course. 

Rescheduling/ Missed Exams: 

With good reason & advanced notification, you may take an exam earlier 
than the scheduled date. If you miss class on the date of the exam, you 
will be required to take a comprehensive exam (conceptual & calculations) 
at the end of the summer session in place of the missed exam - no 
exceptions. If you have not missed an exam, you may take the 
comprehensive exam to replace your lowest exam grade. If you choose 
this option and the comprehensive exam grade is lower than your lowest 
exam grade, the grade will not be averaged with your final grade. 
Absence on an exam date mav be subject to the approval of the Dean of 
the College 

Policy on Cheating / Plagiarism: 

DO NOT CHEAT OR PLAGIARIZE!!! 

Any student caught cheating or plagiarizing will be penalized in 
accordance to the policies stated in the 1999-2000 UNH Student Rights, 
Rules, and Responsibilities. (NO EXCEPTIONS!) 

Students with Disahiiities: 

If you have a disability that requires soecial accommodations, you must 
obtain written documentation from Sv.'fe:' c. 



Course Schedule (All Dates Are Subject to Change!) 
••••• Computer Orientation - June 14"'***** 



Assignment 


Scheduled Date 


Chapters 1-5 


Exam #1 - Monday - June 12* 


Computer Orientation 


Tentatively Scheduled 
Wednesday- June 14* 


Chapters 6-9 


Exam #2 - Wednesday - June 28* 


Work independently 
or in groups on 
computer projects!! 


Monday - July 3'" 


Chapters 10 - 14 


Exam #3 -Monday -July 17* 


Chapters 15, 16, & 18 


Exam #4 - Wednesday - July 26* 


Minitab Projects & 
chapter 19 review 


Presentations - Monday - July 31"* 


Chapter 19 


Conceptual Comprehensive Exam #5 - 
Wednesday - August 2"" 




50 55 60 65 70 75 80 85 90 95 100 
Column 1 






= 77 

^ \2.0S 



2e^\^^ £^^i^^1 



Ch. 15 



1^,6,8. 9. 10, 11, 12, 15, 16 & 17(formula used in class) 



Ch.16 

1, 2, V^l(!dvX^^<^9SiQ<S't<iiS!^^e,<^^) 

Ch.18 
1-3,5.6,8-10,12,13,15,22 



Symbols: 



a 



X^obt 



Fobt 
X^crit 



Fcrit 
fo 






Sw 



Practice Problem #1 

To determine the effect of Ginko-Biloba on short-term memory, an 
experimenter gave a list of 50 words to two groups. One group has 
received Ginko-Biioba, the other received no Ginko-Biloba. Each group is 
allowed to study the list for 5 minutes and then asked to recall as many 
words as possible. The numbers below represent the number of words 
recalled. Use the Mann-Whitney U to evaluate the results (a=.05 i tan). 



Control Group = 



5 
9 
17 
3 



Experimental Group = 



1 

8 

28 

20 

18 



Rank 
Score 


















_ 


Group 




















Rank 





















Procedure for Testing the Null Hypothesis p. . / 

1. State the Null Hypothesis (symtjcis ^^/or words) yj ^.Jt;]^ ^ ^ 

2. State the Alternative Hypothesis (sy misols &/or words) J^r » vv ^^-^ 

directional (1 tailed)} non-directionaiX2 tailed) "^ . /^qj^ 



3. Choose an alpha level / decision rule 

4. Determine the most appropriate statistical analysis 

5. Compute calculations 

6. Make a decision (reject or fail to reject the null hypothesis) 

7. Draw conclusions in tiie context of the problem 









For the following, determine null/alternative, alpha, & the most appropriate statistical 
analysis: 

1 An ecologist suspects that kingbirds found in Switzerland have more feathers than the 
rest of the kingbirds in the riatfefi An exhaustive worldwide study was conducted last 
year to assess the number of feathers on all of the kingbirds in the-aatioa, ^^ <=^'lA 

2. An investigator conducts an experiment to determine the importance of frequency of 
psychotherapy on depression for men and women. Men and woman suffering depression 
are randomly assigned to one of three frequencies of treatment conditions (3X per week, 
IX per week, IX per month). The depression scores are assessed after 6 months. 

3. Prior to the superbowl, a survey was conducted to determine whether there was a 
relationship between gender and team preferences (Tampa Bay Buccaneers or New 
England Patriots). 

4 A health educator wants to evaluate the effect of a dental film on the frequency with 
which children brushed their teeth. Eight children were randomly selected for the 
experiment. First, a baseline of the number of times children brush their teeth in a month 
was established. Next the children are shown the dental film. Again, the numbers of teeth 
brushings are recorded for a month. 

5. A student at Midwest college is interested in whether women or men take more time in 
the shower. 8 women & 8 men are randomly selected to determine weekly shower time. 

6. A traffic safety officer noticed that he was giving more speeding tickets to older people, 
so he conducted an experiment to determine whether there is a relation between people's 
ages & driving speeds. 

7. A professor of women's studies is interested in determining if stress affects the 
menstrual cycle. Ten women are randomly sampled & divided into two groups. One of 
the groups is subjected to high stress for 2 months, while the other group lives in a stress- 
free environment for 2 months. The professor measures the menstrual cycle for all of the 
women. 

8. A researcher believes that women in her tovvTi are taller today than in previous years. 
The researcher compares her data to that of a local consensus collected 20 years ago. 

9. An investigator conducts an experiment to determine the importance of frequency of 
psychotherapy on depression. Subjects suffering depression are randomly assigned to one 
of three frequencies of treatment conditions (3X per week, IX per week, IX per month). 
The depression scores are assessed after 6 months. 

10. During the past 5 years there has been a consistent inflationary trend in milk prices. 
You have yearly average in milk prices for the past year. You are an elementary school 
administrator and need to predict the cost of milk in 2005. 



471 



APPENDIX H 
Descriptive and Comparative Data for PSYC/STAT 104 Class of Summer 2000 
Table HI 



Students ' Expectations & Hopes in Relation to Effort, Grades and Scores, Summer 2000 



Pre: 
Grade 
Hoped 

For? 



Mitch 4 
(II B) 



Pre: 
Grade 

Expected 

in 

course 

(June 5) 



Grade 

in 
Exam 

#1 



Post: 
Post: Grade 
Grade So 
Hoped Far? 
For? (July 
31) 



Time on Prior High 

H'wlc School 

per Highest 

week Math 



Algebraic Final 

Variable Grade in 
Level" PSYC/STAT 
104 



Autumn 4" 
(II Bf 


A 


A 


B^ 

(E=Rr 


A- 


A or 
A^ 


2hrs/wk 


Algll.Disc/Stats 
(Finite Math in 
college) 


4/5 [50] 


A 
(E=R)'' 


Brad 4 
(III A) 


A 


A 


c 

(E>R) 




p/c-] 




Algebra ? 
(Fin 

PSYC/STATS 
in college) 




WP 
(E>R) 


Catherine 
(I) 


A 


A 


A 
(E=R) 


A 


A 


5hrs/wk 


?(Calclin 
college) 


5 [50] 


A 
(E=R) 


Ellen 


Floyd 
(111 A) 


A 


B 


F 
(E>R) 




[F] 




? 




AF 
(E>R) 


Jamie 5 
(II A) 


B 


C 


A 
(E<R) 


C 


B 


5hrs/wk 


Precalculus 

(D* in Psyc 
Stats in college) 


4 [41] 


B* 
(E<R) 


Karen 5 
(III B) 


B 


C 


(E>/= 

R) 


C" 


B 


6-7hrs/wk 


Algebra 11 

(Fin 

PSYC/STATS 
in college) 


2 [26] 


B 
(E<R) 


Kelly 3 
(III B) 


B" 


B" 


F/D" 
(E>R) 




[F/D-] 




Algebra 11 




AF 
(E>R) 


Lee 6 
(II A) 


A 


A 


C 
(E>R) 


B 


B 


20min/wk 


Precalc/calc 
(Finite math, 
college) 


4 [45] 


A" 
(E=R) 



(E>/= 

R) 



3-5hrs/wk Alg I, Geom 4 [43] 
(repeat), 

(Fin 

PSYC/STATS 
in college) 



B 

(E>/=R) 



Mulder 5 
(III A) 


B 


B 


D 

(E>R) 


B 


? 


3hrs/wk Algebra 11 


2 [25] 


B 
(E=R) 


Pierre 8 
(II A) 


A 


B 


(E>R) 


B 


B 


17hrs/wk College prep 


4 [44] 


B^ 
(E=R) 


Robin 3 

(I) 


A 


A 


B" 
(E=R) 


A 


B 


lOhrs/wk "College" 
Alg. (inHS) 




A" 
(E=R) 



Notes: ^Levels of understanding of the algebraic variable on the Algebra Test from the least, 
through 5 the most sophisticated (see Appendix C). The number in the [ ] is the number of items 
correct out of 53. ''Names of individual counseling participants are bolded and the number beside 
their names is the number of their counseling sessions. 'Category Type number (see chapter 7). 
''E=grade expectation, R=grade reality; = less than one grade discrepancy; < or > more than one 
grade discrepancy 



472 



Table H2 



Student Tier (Tobias) and Category (Knowles) in Relation to Class Rank after Exam #1 
and Pre- and Post-Statistical Reasoning Assessment (SRA) scores. 



PRE-Statistics Reasoning 

Assessment [SRA] 

(6/12/2000) 



POST-Statistics Reasoning 

Assessment [SRA] 

(7/31/2000) 



Student in 

order of 

score (Is) 

on pre 

SRA 


Tobias' Tier 

level/Knowles 

Type 


Exam 

#1 

class 

Rank 


Number of 
Is (correct 
reasoning) 


Number of Os 

(misconceptions) 


Number 

of Is 

(correct 

reasoning) 


Number of Os 
(misconceptions) 


Catherine 


1 ^ Tier/Category 1 


V 


13 




4 




13 


(+0) 


4 


(+0) 


Robin 


1" Tier/ Category 1 


3rd 


12 




6 












Kelly 


Unlikely/ Category 
III, type B 


11'" 


11 




6 












Floyd 


Unlikely/ Category 
III, type A/B? 


12'" 


10 




6 












Mulder 


/ Category 
III, type A 


g't> 


9 




3 




7 


(-2) 


6 


(+3) 


Autumn 


Utilitarian/ 
Category II, type B 


4«i> 


9 




4 




9 


(+0) 


4 


(+0) 


Brad 


/ Category III, 
type A 


7'^ 


9 




4 












Mitch 


Utilitarian/ 
Category II, type B 


5,h 


9 




7 




9 


(+0) 


4 


(-3) 


Pierre 


ES0L2°'' Tier/ 
Category II, type A 


gth 


7 




8 




7 


(+0) 


9 


(+1) 


Jamie 


2°" Tier/ Category 
II, type A 


'^nd 


7 




10 




7 


(+0) 


9 


(-1) 


Lee 


2°" Tier/ Category 
II, type A 


6-^ 


6 




9 




7 


(+1) 


8 


(-1) 


Karen 


/ Category III, 
type A 


10'" 


3 




11 




4 


(+1) 


6 


(-5) 






Class 
Average: 






8.75 (n= 


=12) 


6.5 
(n= 


12) 


7.875 
(n=8) 


6.25 
(n=8) 



Notes: ESOL: Pierre was an English Speaker of Other Languages 



473 



Table H3 



Students ' Pre and Post Positions on Feelings and Beliefs Surveys with Net Number of 
Changes 





MATHEMATICS FEELINGS 


MATHEMATICS BELIEFS 






Testing 
Anxiety 


Number 
Anxiety 


Abstraction 
Aaxiety 


Procedural 

to 

Conceptual 


Toxic to 
Healthy 


Learned 
Helpless to 
Mastery 
Oriented 


Performance 
to Learning 
Achievement 
Motivation 


N 
E 
T 


F 

I 

N 
A 
L 


CATEGORY I 

Type A 


















Catherine 


okT**ok 


oktok 


oktok 


oktok 


oktok 


oktok 


oktok 


«; voK 


Robin 


oktnok 


ok=ok 


oktok 


oktok 


oktok 


oktok 


oktok 


+2 


CATEGORY 

II Type A 


















Jamie 


noki*nok 


oki*ok 


noki*ok 


nokinok 


noktnok 


noktnok 


nokt~ok 


+ 1/ 20K 

+4/ 


Lee 


noktnok 


nok^ok 


noktnok 


oktok 


okiok 


ok=ok 


ok=ok 


+5 

+1/ 50K 

-1/ 


Pierre 


oki*ok 


ok=ok 


oktok 


nokt~ok 


okt**ok 


oktok 


nokJ'nok 


+1/ 60K 

+3/ 
+2 


TypeB 


















Autumn 


oktnok 


ok=ok 


oktnok 


nok=nok 


oktok 


noktnok 


noktok 


-21 20K 

0/ 


Mitch 


nokinok 


okJ'ok 


oki**ok 


noktnok 


nokt*ok 


nokinok 


noktnok 


+1 

.1/ 30K 

+4/ 
+5 


CATEGORY 
III 


















Type A 
Brad 


nok 


ok 


nok 


ok 


ok 


ok 


ok 




Frank 


nok 


ok 


ok 


ok 


ok 


ok 


ok 




Karen 


nokt*nok 


nokinok 


nok>l'*ok 


nokt*ok 


oklok 


nok=nok 


okiok 


+21 

+ 1' 40K 



Mulder 


nokinok 


okiok 


ok^lok 


noktok 


okt*ok 


nokt*ok 


oktok 


+2/ 60K 

+6/ 

+7 


TypeB 
Kelly 


nok 


nok 


nok 


nok 


ok 


nok 


nok 





Notes: On scale of 1 through 5: nok: 3.5 to 5 (anxiety) or I to 2.5 (beliefs); nok: 3 to 3.4 (anxiety) or 2.6 
to 3 (beliefs); ok: 2.6 to 3 (anxiety) or 3-3.4 (beliefs); ok: 1 to 2.5 (anxiety) or 3.5-5(beliefs) ; t/i: 
increase/decrease; t*/i*:significant increase/decrease (p < .05);t**/ i**:significant increase/decrease (p < 
.01) using Student t test of difference between means, dependent samples. 



474 



Appendix I 
Summer 2000, PSYC/STAT 104 Class Calendar of Events 

WEEKl 

Interview 1 with Ann Porter May 3 1 , 2000 
Class 1 Wednesday, May 3 1 , 2000, Introductions, the syllabus and schedule, and 
chapter 1 : Statistics and the Scientific Method 

WEEK 2 

Class 2 Monday, June 5, 2000, chapter 2: Basic Mathematical and Measurement 
Concepts and chapter 3: Frequency Distributions. I administered pretest feelings 
and beliefs surveys and invited volunteers to participate in individual mathematics 
counseling. 

STUDY GROUP 1. Wednesday, June 7, 2000 4:30 p.m. 5:45 p.m. 
Riverside Center 
Brad, Lee, Jamie, Pierre (later) 
Class 3. Wednesday, June 7, 2000, chapter 4: Measures of Central Tendency and 
chapter 5: The Normal Curve and Standard Scores 

Individual Sessions. Kelly June 8, 2000 
WEEK 3 

Drop-In. June 12, 2000 Kelly, Karen 

Individual Session. Karen June 12, 2000 
Class 4 Monday, June 12, 2000 Exam 1 on chapters 1 through 5. 1 administered 
the Statistics Reasoning Assessment as a pretest. 

Individual Session. Autumn June 12, 2000 

Brad June 13, 2000 cancelled 
MitchJune 14,2000 
STUDY GROUP 2 Wednesday, June 14, 2000 4:30 p.m. 5:45 p.m. 
Greenville campus in the Learning Assistance Center, Mitch, Lee, 

Kelly 



475 



Class 5 Wednesday, June 14, 2000, Minitab Computer Orientation Computer Lab 
Greenville campus 
Minitab Module 1 

Individual Session. Robin June 14, 2000 

Kelly June 16, 2000 1 1 :30 a.m. 

WEEK 4 

Class 6 Monday, June 19, 2000, chapter 6: Correlation 

Individual Session. Jamie June 20, 2000 5;30 p.m. 
Brad June 20, 2000 6:30 p.m. 
Mulder June 21, 2000 9:30 a.m. 
Lee June 21, 2000 3:20 p.m. 
STUDY GROUP 3 Wednesday, June 21, 2000 
Lee 
Class 7 Wednesday, June 21, 2000, chapter 7: Linear Regression 

Individual Session. Kelly June 21 , 2000 8:20 p.m. 

Pierre June 22, 2000 6:00 p.m.-9:00 p.m. 
Floyd June 23, 2000 9:00 a.m. cancelled 
WEEKS 

Individual Session. Karen Jime 26, 2000 4:00 p.m. 
Mitch June 26, 2000 5:00 p.m. 
Class 8_Monday, June 26, 2000 chapter 8: Random Sampling and Probability and 
chapter 9: Binomial Distribution 

Individual Session. Pierre June 27, 2000 6:30 p.m. 
STUDY GROUP 4 Wednesday June 28, 2000 4:30 p.m. 
Mitch, Lee, Robin, Jamie, Karen (from back of the room). Brad 
(watching). Autumn (with her own questions, Pierre (late), Carol (just 
checking) 
Class 9 Wednesday June 28, 2000, Exam 2 

Individual Session. Autumn June 68, 2000 7:30 p.m. 
Mulder June 29, 2000 8:00 a.m. 



476 



WEEK 6 

Class 10 Monday July 3, 2000, no class meeting 

Individual Session. Jamie July 3, 2000 7:00 p.m. - 8:30 p.m. 
Lee July 5, 2000 3:30 p.m. 
STUDY GROUP 5 Wednesday July 5, 2000 
Lee 
Class 11 Wednesday, July5, 2000, chapter 10: Introduction to Hypothesis testuig 
Using the Sign Test; entirely lecture ...didn't get to Mann Whitney 
Individual Session. Robin July5, 2000 7:30 p.m. 

Mulder July6, 2000 12:00 noon 
WEEK 7 

Interview 2 Ann Porter July 1 0, 2000 3 :00 p.m. 
Individual Session. Karen July 10, 2000 4:00 p.m. 
Class 12 Monday July 10, 2000, chapter 1 1 : Mann- Whitney U Test and chapter 
12: Sampling Distribution of the sample means, the Normal Deviate (z) Test 
Individual Session. Brad July 10, 2000 8:20 p.m. 
Jamie Julyl 1, 2000 10:00 a.m. 
Drop-In Learning Center (with Jillian) 
Lee Julyl2, 2000 1 :00 p.m.- 3:00 p.m. 
Individual Session. Mitch Julyl2, 2000 3:30 p.m. 
STUDY GROUP 6 Wednesday July 12, 2000 
Mitch, Lee 
Class 13 Wednesday Julyl 2, 2000, chapter 13: Student's t Test for Single 
Samples, chapter 14: Student's t Test for Correlated and Independent Groups 
Individual Session. Mulder Julyl2, 2000 8:20 p.m. cancelled 
Pierre Julyl3, 2000 1 1 :00 a.m. 
Brad Julyl3, 2000 6:00 p.m. 
Pierre July 14, 2000 
Robin July 15?, 2000 



477 
WEEKS 

Individual Session Mulder 9:00 a.m. July 17, 2000 
Drop-In : Learning Center (with Jillian) 
Karen 1:00 p.m. -4:00 p.m. 
Jamie didn't come 
Individual Session Karen 4:00 -5 :00 p.m. July 1 7, 2000 

Drop-In : Riverside (with Ann 4:00 p.m. - 6:00 
p.m., with Ann and Jillian 5:00 p.m. - 6:00 p.m.) 
Lee 4:30 p.m. - 6:00 p.m. 
Jamie 

Autumn (doing her own thing) 
Catherine (doing her own work) 
Mitch 5:20 p.m. 

Karen 5:00 p.m. (doing her own thing) 
Class 14 Monday July 17, 2000, Exam 3 

Individual Session Autumn July 17, 2000 7:40 p.m. 
Lee July 19, 2000 3:30 p.m. 
STUDY GROUP 7 Wednesday July 19, 2000 
Lee 
Class 15 Wednesday, July 19, 2000, chapter 15: Introduction to the Analysis of 
Variance chapter 16 Multiple Comparisons, did one-way, talked about setting up 
two-way 

Individual Session Rohm July 1 9, 2000 8:20 p.m. cancelled 
WEEK 9 

Individual Session Karen July24, 2000 4:00 p.m. 
Class 16 Monday July24, 2000, chapter 18: Chi-Square and other Nonparametric 
Tests, namely, one-way and two-way x^ and Wilcoxson Matched-Pairs Test 
Individual Session Mulder July25, 2000 9:00 a.m. 
Robin July25, 2000 ?? 
Pierre July26, 2000 8:00 a.m. 
Jamie July26, 2000 1 0:00 a.m. 
Mitch July26, 2000 3:3 



478 



STUDY GROUP 8 Wednesday July 26, 2000 

Mitch, Lee, Autumn, Jamie, Mulder, Pierre [Robin, Brad, 
Catherine came later] 
Class 17 Wednesday July 26, 2000 Exam 4 

Individual Session Autumn July26, 2000 7:40 p.m. 
WEEK 10 

Drop-In Mulder 1 :00 p.m. 

Individual Session Karen 4:00 p.m. cancelled 
Class 18 Monday July 31, 2000, Minitab Project Presentations, chapter 19: 
Review of Inferential Statistics; I administered research posttests. Brad absent 

Drop-In Karen cancelled 

Individual Session Pierre, August 2, 2000 8:00 p.m. 
Lee, August 2, 2000 3:30 p.m. 

STUDY GROUP 9 Wednesday August 2, 2000 

Lee, Mitch, Autumn, Pierre, Robin, Jamie, Catherine 
Class 19 Wednesday, August 2, 2000 Exam 5, Brad absent 

Individual Session Pierre August 3, 2000 8:00 a.m. 

Interview3 Aim Porter, August 3, 2000, 1 :30 p.m. 
Drop-In: Learning Center (with Jillian) 
Mulder August 3, 2000 (for Finite Math) 
WEEK 11 

Individual Session Jamie August 6, 2000 7:30 p.m. 
Pierre August 7, 2000 8:00 a.m. 
Lee August 7, 2000 9:00 a.m. 



479 



Appendix J 
Sample Mathematics Counselor Tutoring Handouts 




r\ university is considering implementing one of the following three ^^' 

gradmg systems: (1) All grades are pass-fail, (2) all grades are on the 4 

system, and (3) 90% of the grades arc on the 4.0 system and 10% are 

pass-fail. A survey is taken to determine whether there is a relationship 

between undergraduate major and grading system preference. A random 

sample of 200 students with engineering majors, 200 students with arts and 

science majors, and 100 students with fine arts majors is selected Each 

student is asked which of the three grading systems he or she prefers ' 



2 V A physician employed by a large corporation be- 

lieves that due to an increase in sedeniarv life in 
the past decade, middle-age men have become 
fatter. In 1970. the corporation measured the per- 
centage of fat in their employees. For the middle- 
age men. the scores were normally distributed 
with a mean of 22%. To test her hypothesis, the 
physician measures the fat percentage in a ran- 
dom sample of 12 middle-age men currently em- 
ployed by the corporation. The fat percentaces 
found were as follows: 24, 40, 29, 32. 33, 25, 15, 
22. 18. 25, 16. 27. On the basis of these data, can 
we conclude that middle-age men employed by 
the corporation have become fatter? 



^' WdT.r" "^ ' ""tritionis, who has been 

asked to determine whether there is a diff.r^ 



(.. _ A professor has been teaching statistics for many 

^^ years. His records show that the overall mean 

for final exam scores is 82 with a standard devia- 
tion of 10. The professor believes that this year's 
class is superior to his previous ones. The mean 
for final exam scores for this year's class of 65 
students is 87. What do you conclude? 



A 

r^ neuroscientist suspects that low levels of the brain neurotransmitter 
serotonin may be causally related to aggressive behavior As a to s7eD in 

^ -vest.gat.ng this hunch, she decides tf do a correlaUve study TnvoW.^; 

^ . mne rhesus monkeys. The monkeys are observed daily for 6 monZ 2 

the number of aggressive acts recorded. Serotonin levels in the striatum ^a 
bram re^on associated with aggressive behavior) are also measureHi e 



A 



n get 1 additional hour per week in wh.chl 1'*^"'' '^"'^*"8 '""hod 
"nder the guidance of the processor Sint In'' '°"' *""^'^^''^^ P^b'ems 
'"how the methods affecflTems ^ di fen' '"^ 
volunteers for the experiment ZZhLJT^ niathematical abilities 
ab.l.ty info superior, average id poor 't'f 'T''"^ '° mathematical' 
group are randomly assigned fo method I .hh"!' ^'^ ''""^"'^ f™'" «=ach 
to method II. At the end of the coui^e a , 30 f h "'' '™'" ^^'^^ 8^°"P 
exam. The following final exam scor« resuUed '"^ ''"^ ''*= ^^"^^ «-' 

a° ti't,i:ie';"hr^'' 'r '^^'^'^ '-""'''-■ 

-7 "me schizopSi :;end"'"" ''^ ^'"°"'" °^ 

/' director of trainm^ T '" ^" '"«""t'on. As 

aeree.ole h?r. .^ ' ' "^^^''J' institution, you 

mean duration tZi T"' '"^'«"''on. The 

mstaufon is 85 we ks S'"'"""' '''' ^' ^""^ 
«f 15 weeks The^cnr ^ "^""^^'^ "deviation 

The results oI^L? ''' """"""^ distributed, 
'ients.eaedhvrh """"'""' ^""^ '*>=' 'he pa- 

a few years ago) show that the Ebiiam. '"7,""' ^^tires (collected 
Swedish. 8% Irish. 5% German anStrH^,:"^^^^^^ ^"'"^S'^"- ^^^^ 
With percentages under 2% have nm h. f ^^°'^ •''^' nationalities 

random sample of 750 intbtnts" tateVTnd fn'-^ ""^ '"' ^'^ "^'-^ " 
the following table: ''''"• ^"'^ 'he results are shown in 



. A sIcL-p icM.'urclici conilucis an experiment It) 
- determine whethe-r sleep loss affects the ability to 

maintam sustained attention. Fifteen individuals 
are randomly divided into the lollowing three 
groups of 5 subjeos each: group 1. which gets 
t^he normal amount of sleep (7-8 hours); uroup 
C4 2. which is sleep-deprived for 24 hours: and Iiroup 

3. which IS sleep-deprived for 4S hours. Alfthrce 
groups are tested on the same auditory vigilance 
task. Subjects are presented with half-second 
tones spaced at irregular intervals over a I -hour 
duration. Occasionally, one of the tones is slightly 
shorter than the rest. The subject s task "is to 
detect the shorter tones. The following percent- 
ages of correct detections were observed: 



caUntS h'' ""'"''^''^ - •«« for mechani- 
cal ap tude. He wants ,o determine how reliable 

\ O month r °''!' **° admintstrattons spaced by 1 

\ <J month. A study .s conduaed in which 10 students 

are g'ven two administrattons of the test, with 

Ifter the"fi ' rT^'*" "''"^ ^.ven 1 month 
after the first. The data are given in the table. 



A physical education professor believes that ex- 
ercise can slow down the aginu process. For the 
past 10 years, he has been conducting an exercise 
j ^ class for 1 4 individuals wto are currently 50 years 

old. Normally, as one ages, maximum oxygen 
consumption decreases. The national norm for 
maximum oxygen consumption in 50-vear-oid in- 
dividuals is 30 milliliters per kilogram'per minute 
with a standard deviation of 8.6. The mean of 
the 1 4 individuals is 40 milliliters per kilogram per 
minute. What do you condude ' 



T 

mou°nrr'".'"*''"f '°'°"==^^^ g^°""«- the government is considering 
I -? rr ? "^ '^"y-'l^ conservation campaign. However, before doing so 

i Z °"/., " '^"- " ^'"^^ '° =°"'*"" ^" expenment to evaluate the 

effectiveness of the campaign. For the experiment, the conservation cam 
pa.gn IS conducted in a small but representative geographical area. Twelve 
amihes are randomly selected from the area, and the amount of gasoHne 
hey use is monitored for 1 month prior to the advenising campaign and 
for 1 month following the campaign. The following data are coHeCed 



licvos thai ihc amount of smoking by women 
has increased in recent yeais. A complete census 
\ 3- ° ''''^'■'" - ^'^^'■^ -"SO of womtn living m a neigh- 

" ''"''"S ^"y -showed that the mean number of 

cigarettes smol<ed daily by the women was 5.4 
with a standard deviation of 15. To assess 
her belief, the professor determined the daily 
smoking rate of a random sample of :(H) women 
currently living in that city. The data show that 
t^he number of cigarettes smoked daily by the 
200 women has a mean of 6.1 and a standard 
deviation of 2.7. 






A 

Jr^ college professor wants to stetermine the best way to present an 
/ important topic to his class. He has ihe following three choices: ( 1 ) he can 

^5 • lecture, (2) he can lecture plus assign supplementary reading, or (3) he 

can show a film and assign suppiejuentary reading. He decides to do an 
expenment to evaluate the three options. He solicits 27 volunteers from 
his class and randomly assigns 9 to each of three conditions. In condition 
1. he lectures to the students. In comition 2. he lectures plus assigns supple- 
mentary reading. In condition 3, tbe students see a film on the topic plus 
receive the same supplementary reasEng as the students in condition 2. The 
students are subsequently tested oa the material. The following scores 
(percentage correct) were obtained: 



/\ clinical psychologist is interested in the effect that anxietv ha. nn ,h. 

the effect of anxiety depends on the difficulty of the new material An 
/ experiment .s conducted in wh.ch there are three levels o7anxie,vh,fh 

low anx.ou fjl5 " "' '"""''^- °"' «^ ^ P°°' °f volunteers, 15 



'A group of researchers has devised a stress ques- 
t,onna,re cons.stmg of 15 Ufe events T^ey are 
.nterested m determining whether there is Ls^ 
iustrl ^T'"' °" '^" '"'^"^^ ^"'°"n' of ad- 

fndl.H ^^^.^"'^"^^"s and 300 Italians. Each 
•nd.v dual ,s mstructed to use the event of -mar 

required n marnage. Marriage is arbitrarily 
given a value of 50 points. If an event is iudoed 

ve"?rr"" ^'^"""'="' "^^" --^i" '^e 

event should receive more than 50 points How 

adrtrnT''""^'^'''^"'^^""''---hmor: 
adjustment is required. After each subject within 
eac culture has assigned poin. ,o /hTlS liL" 
events, the points for each event are averaged 
The results are shown m the table that follows.' 



APPENDIX K 
Data about Karen 



Table Kl 



Karen's Individual PSYC/STAT 104. Summer 2000 Participation Profile 

Week 



486 



10 



Post 



l"* Class 


Class 


Class 


Class 


Class 


Class 


Class 


Class 


Class 


Class 


Comp 


Mondays 
6:00 p.m. 


2 

Pre- 

Tests 


4 
EXAM 

#1 
JuneI2 


6 


8 


10 

no 

class 

meeting 


12 


14 

EXAM 

#3 
July 17 


16 


18 

Minitab 
Present, 
Post- 
Tests 


(extra) 
EXAM 


2"" Class Class 


Class 


Class 


Class 


Class 


Class 


Class 


Class 


Class 


Class 




Wednesdays ' 
6:00 p.m. 


3 


5 

Minitab 
Partner 
Catherine 
K absent 


7 


9 

EXAM 

#2 
June28 


11 


13 


15 


17 

EXAM 

#4 
July26 


19 

EXAM 

#5 
Aug. 2 




Study Group 

Wednesdays 

4:30 p.m. 








Study 

Gp4 

Kpaitial 














Drop-In 




Jime 










July 


July 


August 

2 
K 

cancelled 








12 










17 


24 








K'/.hr 










K3hrs 


Ky4hr 




Karen's 




June 




June 




July 


July 


July 


July 




Individual 




12 




26 




10 


17 


24 


31 





Sessions 



K 

cancelled 



Meet with 

Instructor 

Extra 



Stud 
Gp. 

before 
EXAM 
w/Jill 5- 

6pm 
IC partial 



487 



Table K2 



Karen's Progress in Tests in Relation to Mathematics Counseling Interventions 




Exam#l 


Exam #2 


Exam #3 


Exam #4 


MINITAB 


Exam 


Optional 




[20% of Final 


[20% of Final 


[20% of Final 


[20% of 


projects 


#5 

1 10% of 

Final 


Exam 




Grade] 


Grade] 


Grade] 


Final 


[10% of 


(to replace 










Grade] 


Final 
Grade] 


Grade] 


lower 
grade) 


General Stra 


egies: average homework 6-7 hours per week, work by se 


f (voluntary) or with experts (me/Ann) in class, 1 


organize, access Individual Mathematics Counseling, and Drop-hi at the Learning Assistance Center | 


Before 


6/12 


6/26 Individual 


7/10 Individual 


7/20 My 


7/10 




1 offered 




Individual 


Math 


Math 


Supervision 


went 




meeting but 




Math 


Counseling 2 


Counseling 1/n 


meeting => 


with /\nn 




Karen 




Counseling 


days before: 


Drop-In: 3hrs + 


have her 


to Comp 




didn't want 




just before 


discussed Exam 


Individual Math 


assess her 


Lab for 




one. 




exam, very 


# 1 . strategy: 


Counseling: Ihr 


own 


help on 








anxious, angry 


enror analysis, 
course strategy 
=> Karen gain 
control, deal 
with math 
depression 


just before: — 
decision flow 
chart + unlabeled 
problems 
=>formula sheet; 
math depression 
lifting 


change, a 

new 

metaphor? 

7/24 

Individual 
Math 
Counseling 


Modi 






Test 


MC:-12,S: 


MC:-I4+2,S: 


MC:-12,S:-1, 


MC:-12,S:- 


Modi: 


96% 


MC: 


Results 


-l,Calc:-25 


-3,Calc:-ll; 


Calc: -2 


0,Calc:-0 


100%; 




12/40 




Total: 62% 


Total: 74% 


Total: 85 + 6% 


Total: 88% 


Present 
with 
Catherine 
Mod 3 
100% 




(30%) 

Calc:45/60 

(75%) 

Total: 57% 


Analysis 


Unhelpful 


Formula Sheet 


Math 


Karen now 










Formula 


issues?; 


Computation 


felt she had 










Sheet; literal 


ALARM: Q6— 


"good enough" 


it in hand 










symbols (N, 


decimals. 


but compared 


(except for 










E) issues 


operations 


"apples' with 


issues with 










statistical 


issues and 


"cheese"; Now 


MC) 










concepts 


literal symbols ; 


has symbols in 












issues 


Language 


hand; Still 












Decimals 


issues-MC and 
Math 


language/ 
concept MC 












fiizzy 


Computation 


issues; 












Preparation 


and S issues; 


Preparation 












issues 


Preparation 
improving 


much improved 










Post 


Individual 


Individual Math 


Individual 


Cancelled 






Will come 


Strategies 


Math 


Counseling: 


Counseling: 


7/31 






to Learning 




Counseling: 


focus: mirror 


focus: provide 


individual 






Center for 




focus: mirror 


her developing 


bearable 


appointment. 






finite 




her embryo 


mathematics 


frustration; 


didn't come 






mathematics 




mathematics 


self; develop JK 


promote growth 


to drop-in 






next 




self; develop 


and Annas 


of realistic self- 


before 






summer "if 




JK and Annas 


secure bases 


esteem, secure 


Exam #5 






Jillian is 




secure bases 




teacher 
attachments 








there" 



KAREN'S Survey Profile Summary 



488 




II 



III 



Not at all 




1 

Not at all 

1.1 1.2 



MATHEMATICS FEELINGS 

Math Testing Anxiety 

3.6 



Number Anxiety 
2.9 



2.8 3 
Abstraction Anxiety 



very much 



1.1 1.5 




T 


1.1 4.5 


'A T 






o 



4.5 



very much 



1,4 


2.8t: 


.0 




A 


A ' 







3.9 very much 
3.5 T 4.2 

-* Ht ► 



1 



a 
2.8 3 



T Pretest Class Extreme 

Scores 

▲ Posttest Extreme Class 

Scores 

T Karen's Pretest Scores 

[placed above the scale] 
a Karen's Posttest Scores 
[placed below scale] 



Figure Kl. Karen's pre and post scores on the pre Mathematics Feelings Survey in 
relation to class extreme scores. 



Mathematics Beliefs Scales 



Procedural Math 



I 



1 



Toxic /Negative 

n 'I 



© 



1 

Lean led Helpless 



KAREN'S Survey Profile Summary 

MATHEMATICS BELIEFS SURVEY 
2.9 



1.9) 2.25 

— " — Jr 



3.55 3.75 
T A- 



3.1 



2.5 2.7 
T A 



3.6 4 

3.9 



-15 — 
3.1 



2.6 



2.1 



■rr- 



a 

2.7 



3. .5 



4 

3.91 



4.4 

— A- 



Process/Conceptual Math 



5 

Healthy/ 
Positive 



5 

Mastery 
Drientated 



T Pretest Class Extreme 

Scores 

A Posttest Extreme Class 

Scores 

T Karen's Pretest Scores 

[placed above the scale] 
a Karen's Posttest Scores 

[placed below scalel 



12 3 4 5 

Figure K2. Karen's pre- and post-summary scores on the Mathematics Beliefs Survey in 
relation to class range scores. 



JMK Mathematics Affect Scales 

1 . When I think about doing mathematics, 

I tend to put work off; 

never 3 



a hi 



sometimes 

2. If I think about how I experience my problems with mathematics, 

I tend to feel discouraged: 

never 5 3 2 very much 

< : > <: [ > 

1 1 4-, 



sometimes 

3. When I think about my mathematics future, 
I feel: 



confident 



'<^ 



=> 
=> 



489 



2 Session #2, June 26, 2000 

3 Session #3, July 10,2000 

4 Session #4, July 17,2000 

5 Session #5, July 24, 2000 
<IlI^ Karen's indication of her 
ranges of response. 

[ ] Karen's written comments 



I feel: 
hopeless/nothing can 

improve 



4. When I think about the mathematics course I am taking now, 
I: 5 



Ilikeil 



<c= 



o 



I: 
"would withdraw if 

I could 



5. When I think about how I do mathematics, 

I: 

feel pride in 2 



how I do It 



4 



:0 



[somewhere 
in here] 



I: 

feel ashamed/ _ 

all the time 



6. When I think of my mathematical achievements, 
I: 2 

feel satisfied 3 



<;= 



^ 



7 While I am doing mathematics. 



I feel 

discouraged 



lean: 
make 



mathematical decisions - 
on my own 



k=^ 



I: 

feel like a complete 

failure/ 



I can." 

not make mathematical 

decisions on my own 



I get 
confijsed 

Figure K3. Karen's responses on the JMK Mathematics Affect Scales^ (in Mathematics 
Counseling Sessions 2 through 5) 



490 



Table K3 



Karen's JMK Mathematics Affect Scales numerical responses. 



JMK Mathematics Affect 


Item 


Item 


Item 


Item 


Item 


Item 


Item 


Average 




Scale 


1 


2 


3 


4 


5 


6 


7 








JMK2:76/26 


0.4 


0.12 


0.5 


0.12 


0.5 


0.5 


0.5 


0.38 


aflE1:62% 


Karen 


JMK3:7/10 


0.5 


0.25 


0.37 


0.25 


0.5 


0.5 


0.5 


0.41 


aflE2:74% 




JMK4;7/17 


0.5 


0.25 


0.5 


0.25 


0.75 


0.5 


0.5 


0.46 


befE3;91% 




JMK5;7/24 


0.5 


0.62 


0.37 


0.3 


0.62 


0.63 


0.5 


0.51 


befE4:88% 






0.48 


0.31 


0.43 


0.23 


0.59 


0.53 


0.53 


0.44 





' See Appendix B for a discussion of the development and rationale for the use of these scales and a copy of 
the survey. 



491 



Table LI 



Appendix L 
Data about Jamie 

Jamie's Individual PSYC/STAT 104. Summer 2000 Participation Profile 

Week 

i ? 3 4 5 6 7 8 9 io~ 



Post 



1^ Class 
Mondays 
6:00 p.m. 


Class 

2 


Class 

4 
EXAM 

1 
Junel2 


Class 

6 


Class 
8 


Class 

10 

no 
class 
meeting 


Class 
12 


Class 

14 
EXAM 

3 
July 17 


Class 
16 


Class 
18 

Minitab 
Post- 
Tests 


Comp 

(extra) 
EXAM 


2"" Class Class 


Class 


Class 


Class 


Class 


Class 


Class 


Class 


Class 


Class 




Wednesda ' 
ys 6:00 
p.m. 


3 


5 

Mini tab 
Partner 
Robin 


7 


9 

EXAM 

2 
June28 


11 


13 


15 


17 

EXAM 

4 
July 26 


19 

EXAM 

5 
Aug. 2 




study Group 

Wednesdays 

4:30 p.m. 


Study 
Gpl 






Study 
Gp4 








Study 
Gp8 


Study 
Gp9 




Drop-In 












No 
Show 










Individual 
Session 






June 
20 

Jillian 
Initiate 




Julys 

Jillian 
Initiate 


July 

11 

Jamie 

initiate 




July 

26 
Jamie 
initiate 




Aug. 
6 

Jamie 
initiate 



Meet with 
Instructor 



Extra 



S.Gp 

before 
EXAM 
w/Ann 

4-6 pm 

w/JUl 5- 

6p-m 



492 



Table L2 



Jamie's Progress in Tests in Relation to Mathematics Counseling Interventions 




Test#l 


Test #2 


Test #3 


Test #4 


MINITAB 
projects 


Tcst#5 


Optional 
Test 




June 12 


June 28 


July 17 


July 26 


July 31 




August 7 




[20% of Final 


[20% of Final 


[20% of Final 


[20% of Final 




August2 






Grade] 


Grade] 


Grade] 


Grade] 


[10% of 

Final 

Grade] 


[10% of 

Final 

Grade] 


(to replace 
lower 
grade) 


General Strate 


gies: average homework 5 hours per week, work by self (involuntary) or with Ann in class (leaves questions unasked, answers | 


unspoken), listen, problem solve. Individual Mathematics Counseling — Jamie initiates last 3 sessions, attend 4 of 9 Snidy Groups (positive if | 


speaks, negatii 


le if not) 






Before 


6/7 Shidv 


6/20 Individual 


7/3 Individual 


My Supervision 




7/26 


8/6 




Group 1:1 


Math Counseling: 


Math 


meetmg => 




Study 


Individual 




asked her a 


analyze Exam# 1 ; 


Counseling Did 


commend her. 




Group 9: 1 


Math 




question. J 


metaphor: inside in 


The Algebra 


give her a 




asked J a 


Counseling 




responded 


storm. 


Test 


bouquet, have 




question. 


Meeting at 




correctly 






her write a new 




Jamie 


Starbucks — 






6/28 Study Group 4: 


7/ II Individual 


metaphor. 




responded 


analyze all 






J watched, listened 


Math 

Counseling 
before t-tests 
covered in class 

7/l7ExtraSmdv 
Group with 
Ann: J watched 
and listened 


7/26 Individual 

Math 

Counseling: 

new metaphor 

Partly sunny day 

1 can come out 

in 

7/26 Study 

Group 8: J 

watched, 

listened 




incorrectly 


Tests. . . see 
her own 
competence 
see changes 
from pre to 
post test 
surveys e.g., 
reduced 
anxiety 


Test Results 


MC:-2,S:-I,C:-2 


MC:-4,S:-5,C:-17; 


MC:-14,S:-0,C- 


MC:-I2,S:-0,C- 


Modi: 


100% 


MC: 23/40 








2 


12 


100%; 




(58%) Calc: 




Total: 95% 


Total: 74% 


Total: 84+ 6% 


Total: 76% 


Present 

with 

Robin 

Mod 

100% 




48/60 (80%) 

Total: 71%: 

Too low to be 
used 


Analysis 


Former 


More anxious than 


Math 


Math 










knowledge plus 


before Test #1 


Computation 


Computation: 










good problem 
solving even 
with panic, 


because of 95% on 
Test tt 1 and family 
pressure; Stady 


"good enough"; 
Now has 
symbols in 


one analysis not 
understood — 
illogical use of 










sound number 


confusing;Formula 


hand; language/ 


literal symbols 










sense 


Sheet issues?; 
Language issues on 
Math Computation, 
and problem solving 
didn't "work'; MC 
"good enough"; 
symbols a problem 


strategy MC 
issues 


cf , numbers, 
one careless fix; 
two logical 
conclusions for 
incorrect calc — 
no credit; MC 
still an issue 








Post 


6/20 Individual 


7/3 Individual Math 


Individual Math 








Probably will 


Strategies 


Math 


Counseling: focus — 


Counseling: 








not go to a 




Counseling: 


Exam Analysis 


focus — 








learning 




focus — 


Counseling 










center for 




commend 












help in 




problem solving 












future; will 
access course 
related group 
help 



493 






fa» 


(1^-l.Y 




































M 






q 




• 










' — - — w 


o iiio 


4 



J N-± 



datrtbuHon. (Spolnti) 






n W TDU Aout Bib MM ■oora* kr 












II 



When Jamie realized lier z in 
question 1 3 was unlikely, she went 
back to question 1 1 , to the s. She 
had found that she had divided the 
sum of squared deviations by 1 00 - 
1, that is, U(- 1, instead of by the 
correct 10-1, that is, « - 1 . When 
she corrected herself, her incorrect 5 
= .63 changed to i' = 2. 1 1 



Figure LI. Jamie's responses on Questions 1 1 and 12 on Exam #1. Note her self- 
corrections in question 1 1 . 



~v 



■■za 


ccnblnflAulQ 


(Be**.) 


-L- 


s 




- 






- 


.3_ 




- 


i«a»i.Mi 





^ 



iXf 



" MWdDMIhaztcoraMirajRadXialy AaitlnAUMIQ.npoMi] 

TKa. 7. score. Htib L((X/-+KajV indl^/iduoul 
& 30+" (X. -fesV" ^corc ixOLj d^i^® oJao/t^ y 

o-i*»ai-\5 TVms in^-vibocul is (XT, ejt(>^vYjlCcif ^ 






When Jamie substituted her 


new 5 = 2. 11 for the 


incorrect 5 = .63 in the z 


formula, the incorrect z of 


4.76 became a more 


reasonable z = 1 .42. 



Figure L2. Jamie's responses on Questions 13 and 14 on Exam #1. Note her self- 
corrections in questionl3. 



494 

Jamie 's Responses on the Algebra Test 

Jamie's level 4 score meant that she was able to treat letters appropriately as 

specific unknowns in some cases, as generalized numbers in some cases, and as variables 

7.(d) 

A = 

( e + 2) 5 




Figure L3. Jamie's solution to question 7 (d) Find the area of the figure. Algebra Test 
(Sokolowski; see Appendix D). Note Jamie's initial error that she scratched out and 
replaced with the correct properly coordinated area solution. 



in others. Jamie's use of letter symbols: 

1. as specific unknowns in some cases (as in Q.14: ...if r = s + t and r + s + t = 30, 
[what is] r?: r = 15 (Jamie's correct response in bold), 

2. as generalized numbers in some cases (as in Q.I8: When [is this] true...? L + M + 
N = L + P + N Always Never Sometimes , when: M =P (Jamie's correct 
response in bold), and 

3. as variables in others (as in 5. (c) If e + f = 8 then e + f + g =: 8 + g (Jamie's 
correct response in bold). (Appendix C, Algebra Test) 

In addition, she was able to resolve ambiguity by coordinating two operations. For 
example, to determine the area of a rectangular figure she corrected her initial impulse to 
incorrectly use only one operation, multiplication, to get lOe, to the coordination of 
addition and multiplication, to obtain (e + 2)5 (see Figure L3). 



495 



JAMIE'S Survey Profile Summary 

MATHEMATICS FEELINGS 
Math Testing Anxiety Pre 4.1 



Not at all 
1.1 



'*-r 



1.5 



T 

4.1 



3 

Post 3.6 



Abstraction Anxiety 



Not at all 

1.1 1.2 
ffl 



Pre 3.7 



••^T- 



3.5 

—r 



Post 3.0 



4.2 



very much 



4.5 

s. — 



Not£ 


tall 


P 

1.4 


re 2.1 

T 


Numbe 
2.. 1 


r Anxiety 


veryn 


11 




T (S 

A Post 1.5 




•1 





very much 

►■ 



▼ Pretest Class Extreme Scores 

A Posttest Extreme Class 

Scores 

T Jamie's Pretest Scores 

[placed above the scale] 
a Jamie's Posttest Scores 
[placed below scale] 



MATHEMATICS BELIEFS SURVEY 



Process/Relational Math 





1.9 


I '■'! 


T 


3.55 3.75 






1 
Tox 


1 : 
ic /Negative 


I 


Post2.6 
Pre2.5 

T 
2.5 2.7 


T A 

3.9 


Healthy 

J 4.4 


5 
/Positive 


n 






T ▲ 




' A ' 




Lear 


1 : 

Pr 

ned Helpless t 

2 


I 

b2 

2.1 


Post2.7 


5 -! 

3.5 3.9 


Masteiy 


5 
Orientated 


in 


Post 


A 
2.1 




T A 







Figure L4. Jamie's mathematics pre- and post-feelings and beliefs Survey Profile 
Summary in relation to class range pre-and post-scores 

Session 1. We discussed briefly Jamie's anxiety average scores on the Mathematics 
Feelings pretests that I had plotted with the class extreme scores on her Survey Profile 
Summary (see Figure L4). Jamie's Number Anxiety was low (2. 1), close to the middle of ■ 
the class range, but her Abstraction Anxiety was high. She was not surprised by her high 
Math Testing Anxiety score (the highest in the class at 4.1). 



JIVIK Mathematics Affect Scales 



496 



1 . When I think about doing mathematics, 
I tend to put work off: 
never 2 

1 3 
S 4, 1 



a lot 



sometiities 

2. If I think about how I experience my problems with mathematics, 
1 tend to feel discouraged: 

never 12 3 verv much 

5 4 



sometimes 

3. When I think about my mathematics future, 
Ifeel: 

1 



confident 



1 Session 1, June 20, 2000 

2 Session 2, July 3, 2000 

3 Session 3, July 11,2000 

4 Session 4, July 26, 2000 

5 Session 5, August 6, 2000 



Ifeel: 
hopeless/nothing can 

improve 



4. When I think about the mathematics course I am taking now, 

I: 1: 

would withdraw if 



I like it 



4 2 

5 3 



5. When I think about how I do mathematics, 
I: 



feel pride in 
how I do it 



1 5 4 



could 



1: 

feel ashamed/_ 

all the lime 



6. When I think of my mathematical achievements. 



feel satisfied 4 






5 


2 


7. W 
1 can: 
make 


Ifeel 
discouraged 
hile I am doing mathematics. 


mathematical decisions 




on my 


own 4 1 


2 



I: 

feel like a complete 

fa lure/ 



I can." 

not make mathematical 

decisions on my own 



I get 
Confused 



Figure L5. Jamie's responses on the JMK Mathematics Affect Scales, Sessions 1-5 



497 



Table L3. Numerical scores and averages of Jamie's JMK Scale responses (see Figure K5) 



JMK Mathematics Affect 


Item 


Item 


Item 


Item 


Item 


Item 


Item 


Average 




Scale 


1 


2 


3 


4 


5 


6 


7 








JMK1:6/20 


0.87 


0.75 


0.37 


0.9 


0.82 


1 


0.75 


0.78 


aftE1:95 


"Jamie" 


JMK2:7/3 


0.75 


0.5 


0.37 


0.63 


0.5 


0.7 


0.5 


0.564286 


anE2;74 




JMK3:7/11 


0.75 


0.25 


0.37 


0.63 


0.37 


0.4 


0.5 


0467143 


befE3:90 




JMK4:7/26 


0.75 


0.5 


0.5 


0.75 


0.7 


0.75 


0.75 


0.671429 


befE4:76 




JMK5:8/6 


0.87 


0.75 


0.63 


0.75 


0.75 


0.75 


0.75 


0.75 


befEC:71 






0.80 


0.55 


0.45 


0.732 


0.63 


0.72 


0.65 


0.646571 





498 



APPENDIX M 

Data about MULDER 
Table Ml. 
Mulder's Individual PSYC/STAT 104, Summer 2000 Participation Profile 

Week 



10 



Post 



1^ Class 


Class 


Class 


Class 


Class 


Class 


Class 


Class 


Class 


Class 


Mondays 
6:00 p.m. 


2 
Pre- 
Tests 


4 

EXAM 

#1 

Junel 

2 

M 

late 


6 


8 


10 

no 
class 

meeting 


12 


14 

EXAM 
#3 

July 
17 


16 


18 

MINITAB i 

Post- 
Tests 


2"" Class Class 


Class 


Class 


Class 


Class 


Class 


Class 


Class 


Class 


Class 


Wednesdays 1 
6:00 p.m 


3 


5 

Minitab 
Partner 
Pieiic 


7 


9 

EXAM 

#2 
June28 


11 


13 


15 


17 
EXAM 

#4 
July 26 


19 

EXAM 

#5 
Aug. 2 


study Group 

Wednesdays 

4:30 p.m. 
















Study 
Gp8 




Drop-In 


















July 31 

Minitab 

paper, 

Aug. 3 

MATH 

120 

Finite 


Individual 






June 


June 


July 


July 


July 


July 




Session 






21 


29 


6 


12 

M 


17 


25 










Stats 


Finite 

math& 

Stats 


Stats 


cancelled 


Stats 


Stats 










only 


only 




only 


only 





Meet with 
Instructor 

Extra 



Gp 

before 

EXAM 
w/Ann 

4-6 p m 

w/Jill 5- 

6 p.m 



Table M2 

Mulder 's Progress in Tests in Relation to Mathematics Counseling Interventions 



499 





Test#l 


Test #2 


Test #3 


Test #4 


MINITAB 


Test#5 


Test 




6/12/00 


6/28/00 


7/17/00 


7/26/00 


projects 
7/31/00 


8/2/00 
[10% of 


(10 

replace 




[20% of Final 


[20% of Final 


[20% of Final 


[20% of Final 


[10% of 


Final 


lower 
grade) 




Grade] 


Grade] 


Grade] 


Grade] 


Final 


Grade] 












Grade] 






General Strat 


egies: average homework 3 hours per 


week, work with others in class, visualize/memorize access 




Individual M 


athematics Couns 


eling, and Drop-In at the Learning Assistance Center 








Before 




6/21 First 


7/6 Individual 


My 


7/31 










Individual 


Math 


Supervision 


dropped 










Math 


Counseling: 


meeting=> 


in to have 










Counseling; 


Finite Math 


trial MC 


me read 










analysis of 


7/17 9:00am 


resistance Test 


his and 










Exam# 1 -lack of 


Individual 


at 7/25 


Pierre's 










prep, lack of 


Math 


Individual 


MINITAB 










knowing what 


Counseling 


Math 


project 










to expect 


focused on 
Exam #3; focus 
on symbol 
links; choosing 
and doing 
hypothesis test 


Counseling 


write up 






Test 


MC:-18,S:- 


MC:-14,S:- 


MC:-I8,S:- 


MC:-8+2,S:- 


Mod: 


94% 




Results 


l,Calc:-18 


5,Calc:-0 


0,Calc:-6 


0,Calc:-3 


100%; 








Total: 63 


Total: 81 


Total: 76 + 5 


Total: 91 


Present 

with 

Pierre 

Mod 

92% 






Analysis 


Poor Formula 


Has Math 


Math 


Mulder has 










Sheet; MC 


Computation 


Computation 


mastered the 










issues; lack of 


more in hand; 


still OK; Now 


last hurdle: 










study 


Verbal-MC. S 
issues 


has symbols in 
hand; STE.L 
big MC issues 


MC! 








Post 


Individual 


6/29 Individual 


Individual 






Drop-in 


Probably 


Strategies 


Math 


Math 


Math 






help with 

Finite 

Word 


won t 
goto 




Counseling; 


Counseling: 


Counseling: 






a 




focus — overall 


[half on finite 


focus MC, 






Problems 


LcurnjQg 

Center 




approach; 


math: simplex 


resistance 








when 




Formula Sheet 


method] 
focus — verbal 
connections 
especially 
symbols 










taking 
math 
in 
fiiture 



Not at all 



I* 



MULDER'S Survey Profile Summary 
MATHEMATICS FEELINGS 

Math Testing Anxiety 

3.1 very much 



1.1 1.5 




' 


4.1 4.5 1 


< A T 


i! 


' A ' 



1 2 

Number Anxiety 
Not at all 2.1 



n 



1 2 

Abstraction ArLxiety 
Not at all 



very much 



1.4 


T 2.8 


h« 




^ T a 

▲ 1.8 


M 





2.7 



T Pretest Class 

Extreme Scores 

APosttest 

Extreme Class 

Scores 

X Mulder's 

Pretest Scores 

[placed above 
the scale] 



1.1 1,2 


T 


3.5 


4.2 


^T A c 

1.9 




A 


T * 



veo'much ^ lyiuijer's 



111 



1 2 3 

MATHEMATICS BELIEFS SURVEY 
Procedural Math 2.9 



3.051 



Posttest Scores 

[placed below 
scale] 



1.9 


i 2.25 X 


3.55 3.75 




^ ■f 


A 


a T A 





1 

Toxic /Negative 



3.2 



II 



2.5 2.7 

y A 



4 

39B 



Process/Relational Math 



Healthy/Positive 



1 2 

^ — -Learned Helpless 



2.6 



(^ 



2 2.1 



"S — 

3.5 



3.5 



4 

3.91 



4.4 



Mastery Orientated 



500 



Figure Ml. Mulder's responses to the pre- and post-Mathematics Feelings and 
Mathematics Beliefs surveys in relation to class extreme scores. 



JMK Mathematics Aifect Scales 



1 . When I think about doing mathematics, 

I tend to put work off: 

never 



a hi 
1 



501 



I Sessional, June 21 2000 

3 Session #3, July 6, 2000 
5 Session #5, July 25, 2000 



2. If I think about how I experience my problems with mathematics. 



I tend to feel discouraged: 
never 3 



very much 



3. When I think about my mathematics future, 
I feel: 

confident 1 

3 

5 



I feel: 

hopeless/nothing can 
improve 



When I think about the mathematics course I am taking now. 



/ like it 



5. When I think about how 1 do mathematics, 

1: 

feel pride in 

how I do it 5 3 1 



6. When I think of my mathematical achievements, 

I: 1 

feel satisfied 3 

5 



7. While 1 am doing mathematics, 

lean: 

make 

mathematical decisions 
on my own 

5 3 



I feel 
discouraged 



Iget 
confused 



would withdraw if 
could 



feel ashamed/ _ 
all the time 



feel like a complete 
fa lure/ 



lean; 

not make mathematical 

decisions on mv own 



Figure M2. Mulder's responses on the JMK Mathematics Affect Scales for Sessions 1,3, 
and 5. 



502 



Table M3. 

Numerical Scores and Averages of Muldefs JMK Affect Scale Responses 



JMK Mathematics 


Item 


Item 


Item 


Item 


Item 


Item 


Item 


Average 




Affect Scale 


1 


2 


3 


4 


5 


6 


7 










JMK1:6/21 


0.0 


0.5 


075 


0.75 


0.5 


0.75 


0.60 


0.55 


aftE1:63' 


Mulder 


JMK3:7/6 


0.25 


0.75 


0.75 


0.75 


0.63 


0.75 


0.60 


0.64 


aftE2:81befE3:81 




JMKS:7/25 


0.5 


0.5 


0.75 


0.75 


0.75 


0.75 


0.75 


0.68 


befE4:91'= 




Average: 


0.34 


0.58 


0.75 


0.75 


0.62 


0.75 


0.68 


0.64 







Notes: ^ after Exam #1 where he earned 63%; ''after Exam #2 where he earned 81% and 
before Exam #3 where he earned 81% (with extra credit); '^ after Exam #4 where he earned 
91%. 



503 
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